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Estimation

linear-demand linear-cost, \(\sigma = 1\)

\(\theta = 0.05\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.817 1.000 7.959 9.580 15.288
alpha1_hat 100 0 0.825 0.857 −1.470 0.766 3.413
alpha2_hat 100 0 0.110 0.081 −0.105 0.109 0.348
alpha3_hat 100 0 0.979 0.217 0.494 0.962 1.691
gamma0_hat 100 0 3.077 0.886 0.820 3.074 5.536
gamma1_hat 100 0 1.499 6.355 −13.374 0.682 56.754
gamma2_hat 100 0 0.774 0.141 0.455 0.774 1.072
gamma3_hat 100 0 0.780 0.169 0.438 0.771 1.227
theta_hat 100 0 −0.654 3.777 −33.399 −0.165 9.249
theta_hat_t_stats 100 0 −0.418 1.124 −3.264 −0.307 2.625
R2_demand 100 0 0.370 0.292 −1.623 0.433 0.715
R2_supply 100 0 0.983 0.004 0.970 0.984 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.978 0.748 8.375 9.847 12.592
alpha1_hat 100 0 0.829 0.670 −0.830 0.745 2.707
alpha2_hat 100 0 0.116 0.061 −0.056 0.113 0.304
alpha3_hat 100 0 0.999 0.167 0.570 0.990 1.613
gamma0_hat 100 0 2.139 0.901 −0.193 2.333 4.099
gamma1_hat 100 0 0.658 1.834 −11.212 0.939 4.612
gamma2_hat 100 0 0.872 0.144 0.545 0.852 1.269
gamma3_hat 100 0 0.873 0.132 0.563 0.859 1.217
theta_hat 100 0 0.024 0.892 −1.713 −0.073 5.346
theta_hat_t_stats 100 0 −0.171 0.971 −2.186 −0.158 2.515
R2_demand 100 0 0.325 0.241 −0.787 0.387 0.644
R2_supply 100 0 0.980 0.004 0.969 0.980 0.988
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.973 0.410 9.143 9.948 10.873
alpha1_hat 100 0 0.947 0.358 0.030 0.932 1.842
alpha2_hat 100 0 0.104 0.034 0.028 0.103 0.204
alpha3_hat 100 0 0.998 0.103 0.757 0.985 1.293
gamma0_hat 100 0 1.511 0.676 −0.463 1.639 2.851
gamma1_hat 100 0 0.865 0.907 −3.144 0.848 3.198
gamma2_hat 100 0 0.938 0.099 0.746 0.931 1.208
gamma3_hat 100 0 0.946 0.094 0.760 0.932 1.192
theta_hat 100 0 0.029 0.459 −1.402 0.076 1.969
theta_hat_t_stats 100 0 0.048 1.056 −3.335 0.239 2.741
R2_demand 100 0 0.332 0.127 −0.047 0.341 0.540
R2_supply 100 0 0.977 0.004 0.965 0.978 0.986
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.997 0.199 9.646 9.993 10.439
alpha1_hat 100 0 0.995 0.169 0.621 0.983 1.359
alpha2_hat 100 0 0.101 0.018 0.067 0.101 0.140
alpha3_hat 100 0 1.001 0.046 0.901 0.994 1.106
gamma0_hat 100 0 1.126 0.300 0.300 1.142 1.798
gamma1_hat 100 0 0.929 0.304 0.177 0.931 1.909
gamma2_hat 100 0 0.986 0.049 0.881 0.979 1.124
gamma3_hat 100 0 0.987 0.041 0.868 0.986 1.096
theta_hat 100 0 0.064 0.160 −0.460 0.068 0.510
theta_hat_t_stats 100 0 0.363 0.930 −2.190 0.402 2.791
R2_demand 100 0 0.342 0.050 0.237 0.340 0.458
R2_supply 100 0 0.975 0.002 0.970 0.975 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.146 9.640 9.993 10.428
alpha1_hat 100 0 1.012 0.118 0.683 1.016 1.274
alpha2_hat 100 0 0.099 0.011 0.073 0.099 0.124
alpha3_hat 100 0 1.000 0.037 0.909 1.002 1.081
gamma0_hat 100 0 1.095 0.203 0.589 1.098 1.581
gamma1_hat 100 0 0.926 0.242 0.351 0.933 1.440
gamma2_hat 100 0 0.989 0.031 0.913 0.989 1.076
gamma3_hat 100 0 0.993 0.026 0.925 0.995 1.053
theta_hat 100 0 0.072 0.129 −0.197 0.065 0.334
theta_hat_t_stats 100 0 0.604 1.080 −1.715 0.565 2.765
R2_demand 100 0 0.332 0.041 0.221 0.332 0.431
R2_supply 100 0 0.975 0.001 0.972 0.975 0.978
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.080 9.808 10.004 10.179
alpha1_hat 100 0 1.000 0.075 0.819 0.995 1.214
alpha2_hat 100 0 0.100 0.007 0.080 0.101 0.117
alpha3_hat 100 0 1.000 0.021 0.950 0.999 1.063
gamma0_hat 100 0 1.024 0.153 0.632 1.037 1.393
gamma1_hat 100 0 0.916 0.131 0.542 0.936 1.200
gamma2_hat 100 0 0.997 0.021 0.936 0.996 1.043
gamma3_hat 100 0 0.999 0.020 0.961 0.997 1.057
theta_hat 100 0 0.089 0.073 −0.067 0.082 0.319
theta_hat_t_stats 100 0 1.185 0.961 −0.962 1.132 4.005
R2_demand 100 0 0.336 0.022 0.287 0.336 0.389
R2_supply 100 0 0.975 0.001 0.972 0.975 0.977

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.059 1.165 7.650 9.738 14.910
alpha1_hat 100 0 1.176 1.935 −3.982 1.278 7.462
alpha2_hat 100 0 0.490 0.207 0.124 0.465 1.534
alpha3_hat 100 0 0.985 0.246 0.477 0.942 1.737
gamma0_hat 100 0 3.015 0.997 0.487 3.036 5.063
gamma1_hat 100 0 0.263 2.408 −5.489 0.308 8.755
gamma2_hat 100 0 0.781 0.175 0.384 0.780 1.432
gamma3_hat 100 0 0.780 0.159 0.356 0.787 1.298
theta_hat 100 0 −0.101 0.418 −1.075 −0.126 0.836
theta_hat_t_stats 100 0 −0.397 1.096 −3.247 −0.373 1.804
R2_demand 100 0 0.458 0.301 −1.023 0.537 0.783
R2_supply 100 0 0.977 0.005 0.961 0.977 0.989
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.954 0.651 8.636 9.896 11.753
alpha1_hat 100 0 1.072 1.230 −1.860 1.047 3.562
alpha2_hat 100 0 0.486 0.131 0.170 0.479 0.855
alpha3_hat 100 0 1.000 0.157 0.639 0.996 1.408
gamma0_hat 100 0 2.018 0.744 −0.096 2.122 3.772
gamma1_hat 100 0 0.610 1.432 −3.065 0.705 5.362
gamma2_hat 100 0 0.891 0.113 0.599 0.894 1.243
gamma3_hat 100 0 0.886 0.121 0.583 0.890 1.197
theta_hat 100 0 −0.023 0.257 −0.936 −0.033 0.540
theta_hat_t_stats 100 0 −0.168 0.971 −2.766 −0.150 1.607
R2_demand 100 0 0.513 0.142 0.085 0.535 0.744
R2_supply 100 0 0.972 0.005 0.956 0.972 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.044 0.566 8.881 10.032 11.537
alpha1_hat 100 0 1.174 1.117 −1.140 0.994 4.438
alpha2_hat 100 0 0.489 0.121 0.227 0.496 0.754
alpha3_hat 100 0 1.002 0.118 0.800 0.995 1.399
gamma0_hat 100 0 1.610 0.638 −0.109 1.650 3.243
gamma1_hat 100 0 0.719 1.094 −2.132 0.852 3.085
gamma2_hat 100 0 0.934 0.101 0.679 0.927 1.175
gamma3_hat 100 0 0.937 0.098 0.719 0.927 1.223
theta_hat 100 0 0.025 0.198 −0.358 −0.010 0.611
theta_hat_t_stats 100 0 0.051 0.949 −1.910 −0.051 2.024
R2_demand 100 0 0.510 0.116 0.133 0.530 0.706
R2_supply 100 0 0.969 0.004 0.958 0.970 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.175 9.584 10.020 10.482
alpha1_hat 100 0 1.003 0.360 0.058 0.995 1.761
alpha2_hat 100 0 0.501 0.041 0.418 0.497 0.596
alpha3_hat 100 0 1.004 0.044 0.928 1.001 1.117
gamma0_hat 100 0 1.096 0.269 0.427 1.086 1.873
gamma1_hat 100 0 0.853 0.438 −0.121 0.881 1.988
gamma2_hat 100 0 0.988 0.042 0.869 0.986 1.111
gamma3_hat 100 0 0.985 0.043 0.870 0.985 1.112
theta_hat 100 0 0.062 0.084 −0.182 0.059 0.242
theta_hat_t_stats 100 0 0.720 0.982 −2.081 0.754 2.786
R2_demand 100 0 0.519 0.039 0.418 0.519 0.600
R2_supply 100 0 0.965 0.002 0.959 0.965 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 0.137 9.681 9.998 10.394
alpha1_hat 100 0 0.991 0.228 0.396 1.004 1.556
alpha2_hat 100 0 0.503 0.027 0.432 0.503 0.568
alpha3_hat 100 0 1.005 0.031 0.929 1.009 1.097
gamma0_hat 100 0 1.048 0.224 0.376 1.037 1.658
gamma1_hat 100 0 0.842 0.349 −0.032 0.874 1.637
gamma2_hat 100 0 0.995 0.035 0.918 0.995 1.118
gamma3_hat 100 0 0.994 0.034 0.925 0.989 1.088
theta_hat 100 0 0.069 0.063 −0.059 0.067 0.214
theta_hat_t_stats 100 0 1.171 1.054 −1.040 1.144 3.476
R2_demand 100 0 0.520 0.031 0.432 0.520 0.580
R2_supply 100 0 0.965 0.002 0.960 0.965 0.969
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.089 9.823 10.002 10.274
alpha1_hat 100 0 1.009 0.192 0.503 1.023 1.425
alpha2_hat 100 0 0.500 0.019 0.454 0.501 0.557
alpha3_hat 100 0 0.999 0.019 0.947 1.001 1.035
gamma0_hat 100 0 0.995 0.118 0.764 0.989 1.319
gamma1_hat 100 0 0.931 0.182 0.274 0.926 1.300
gamma2_hat 100 0 1.000 0.018 0.957 1.001 1.041
gamma3_hat 100 0 1.001 0.020 0.956 1.001 1.045
theta_hat 100 0 0.063 0.035 −0.025 0.067 0.164
theta_hat_t_stats 100 0 1.706 0.945 −0.710 1.807 4.355
R2_demand 100 0 0.519 0.021 0.463 0.522 0.557
R2_supply 100 0 0.964 0.001 0.962 0.964 0.967

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.933 0.897 7.453 9.922 12.191
alpha1_hat 100 0 0.943 3.335 −7.741 1.125 8.999
alpha2_hat 100 0 0.990 0.372 −0.355 0.966 1.926
alpha3_hat 100 0 0.992 0.218 0.563 0.996 1.477
gamma0_hat 100 0 2.836 0.960 0.079 2.991 5.005
gamma1_hat 100 0 0.995 3.533 −11.836 0.886 9.696
gamma2_hat 100 0 0.809 0.159 0.481 0.789 1.310
gamma3_hat 100 0 0.812 0.169 0.502 0.791 1.429
theta_hat 100 0 −0.187 0.402 −1.684 −0.186 1.548
theta_hat_t_stats 100 0 −0.710 1.124 −4.112 −0.585 1.549
R2_demand 100 0 0.599 0.163 −0.154 0.641 0.834
R2_supply 100 0 0.973 0.007 0.946 0.974 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.640 8.767 9.912 11.649
alpha1_hat 100 0 1.277 2.296 −3.572 1.307 9.090
alpha2_hat 100 0 0.957 0.243 0.171 0.962 1.500
alpha3_hat 100 0 0.996 0.141 0.591 0.976 1.348
gamma0_hat 100 0 2.022 0.833 −0.589 2.029 4.037
gamma1_hat 100 0 0.160 2.450 −8.517 0.256 6.340
gamma2_hat 100 0 0.888 0.129 0.592 0.864 1.272
gamma3_hat 100 0 0.868 0.128 0.570 0.876 1.183
theta_hat 100 0 −0.005 0.260 −0.597 −0.025 0.806
theta_hat_t_stats 100 0 −0.130 1.020 −2.309 −0.104 2.600
R2_demand 100 0 0.562 0.128 0.062 0.583 0.757
R2_supply 100 0 0.968 0.007 0.935 0.969 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.499 8.849 9.898 12.162
alpha1_hat 100 0 1.156 1.481 −2.597 1.140 5.041
alpha2_hat 100 0 0.986 0.184 0.567 0.987 1.489
alpha3_hat 100 0 0.997 0.102 0.805 0.980 1.371
gamma0_hat 100 0 1.551 0.630 −0.070 1.570 3.106
gamma1_hat 100 0 0.604 1.457 −3.604 0.518 4.245
gamma2_hat 100 0 0.945 0.093 0.749 0.944 1.192
gamma3_hat 100 0 0.942 0.095 0.750 0.930 1.313
theta_hat 100 0 0.013 0.147 −0.369 0.009 0.470
theta_hat_t_stats 100 0 0.056 0.882 −1.815 0.049 2.129
R2_demand 100 0 0.570 0.128 −0.289 0.581 0.734
R2_supply 100 0 0.965 0.006 0.946 0.966 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.006 0.190 9.664 10.007 10.476
alpha1_hat 100 0 1.074 0.713 −0.916 1.075 3.064
alpha2_hat 100 0 0.994 0.084 0.760 0.989 1.234
alpha3_hat 100 0 0.998 0.048 0.896 0.995 1.125
gamma0_hat 100 0 1.123 0.281 0.222 1.122 1.709
gamma1_hat 100 0 0.666 0.653 −1.456 0.676 2.163
gamma2_hat 100 0 0.985 0.042 0.868 0.982 1.096
gamma3_hat 100 0 0.997 0.040 0.909 0.996 1.103
theta_hat 100 0 0.067 0.071 −0.130 0.067 0.317
theta_hat_t_stats 100 0 0.882 0.950 −1.893 0.893 3.821
R2_demand 100 0 0.570 0.032 0.490 0.574 0.662
R2_supply 100 0 0.961 0.003 0.954 0.961 0.968
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.010 0.126 9.731 9.995 10.335
alpha1_hat 100 0 1.019 0.463 0.015 1.021 2.032
alpha2_hat 100 0 1.000 0.053 0.850 1.000 1.129
alpha3_hat 100 0 1.003 0.031 0.938 1.005 1.062
gamma0_hat 100 0 1.020 0.215 0.530 1.013 1.578
gamma1_hat 100 0 0.769 0.515 −0.381 0.763 1.804
gamma2_hat 100 0 1.000 0.031 0.917 1.004 1.063
gamma3_hat 100 0 0.995 0.029 0.909 0.992 1.080
theta_hat 100 0 0.067 0.049 −0.042 0.063 0.183
theta_hat_t_stats 100 0 1.295 0.942 −0.858 1.253 3.598
R2_demand 100 0 0.571 0.024 0.493 0.572 0.614
R2_supply 100 0 0.960 0.002 0.953 0.960 0.966
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 0.085 9.768 10.013 10.200
alpha1_hat 100 0 1.054 0.308 0.219 1.050 1.716
alpha2_hat 100 0 0.997 0.033 0.930 0.993 1.075
alpha3_hat 100 0 1.003 0.020 0.959 1.002 1.049
gamma0_hat 100 0 1.010 0.144 0.637 1.012 1.378
gamma1_hat 100 0 0.878 0.322 −0.008 0.889 1.800
gamma2_hat 100 0 1.000 0.020 0.952 1.000 1.052
gamma3_hat 100 0 0.998 0.022 0.933 0.999 1.050
theta_hat 100 0 0.059 0.034 −0.026 0.059 0.142
theta_hat_t_stats 100 0 1.835 1.015 −0.792 1.819 4.317
R2_demand 100 0 0.567 0.017 0.533 0.569 0.607
R2_supply 100 0 0.960 0.002 0.956 0.960 0.963

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.027 0.973 8.184 9.967 14.140
alpha1_hat 100 0 2.793 15.115 −34.879 2.222 36.412
alpha2_hat 100 0 4.848 1.764 1.841 4.817 9.933
alpha3_hat 100 0 1.004 0.222 0.481 0.970 1.743
gamma0_hat 100 0 3.089 1.022 0.466 3.230 5.484
gamma1_hat 100 0 1.026 13.342 −28.423 1.896 42.252
gamma2_hat 100 0 0.747 0.162 0.449 0.725 1.179
gamma3_hat 100 0 0.786 0.155 0.409 0.784 1.168
theta_hat 100 0 −0.190 0.297 −1.024 −0.181 0.618
theta_hat_t_stats 100 0 −0.794 1.045 −3.431 −0.842 1.642
R2_demand 100 0 0.591 0.189 −0.415 0.620 0.827
R2_supply 100 0 0.970 0.008 0.949 0.971 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.961 0.545 8.679 9.945 11.240
alpha1_hat 100 0 1.358 8.303 −18.125 1.695 18.536
alpha2_hat 100 0 4.939 0.965 2.676 4.963 7.482
alpha3_hat 100 0 0.994 0.134 0.628 0.996 1.392
gamma0_hat 100 0 2.042 0.695 0.286 2.186 3.414
gamma1_hat 100 0 −3.602 9.247 −31.453 −2.574 19.678
gamma2_hat 100 0 0.887 0.107 0.612 0.872 1.138
gamma3_hat 100 0 0.890 0.124 0.547 0.885 1.241
theta_hat 100 0 0.020 0.208 −0.449 0.004 0.722
theta_hat_t_stats 100 0 0.013 0.985 −2.259 0.024 2.894
R2_demand 100 0 0.622 0.105 0.291 0.640 0.816
R2_supply 100 0 0.965 0.008 0.945 0.966 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.967 0.396 9.046 9.941 10.802
alpha1_hat 100 0 0.952 7.211 −15.549 1.057 18.070
alpha2_hat 100 0 4.972 0.807 3.013 4.915 6.846
alpha3_hat 100 0 1.001 0.099 0.769 0.993 1.269
gamma0_hat 100 0 1.581 0.645 −0.328 1.640 2.854
gamma1_hat 100 0 −0.651 7.033 −18.899 −0.772 19.644
gamma2_hat 100 0 0.933 0.096 0.721 0.923 1.289
gamma3_hat 100 0 0.942 0.101 0.699 0.921 1.196
theta_hat 100 0 0.018 0.164 −0.444 0.009 0.444
theta_hat_t_stats 100 0 0.014 1.118 −3.397 0.077 2.303
R2_demand 100 0 0.620 0.068 0.439 0.633 0.746
R2_supply 100 0 0.960 0.007 0.937 0.962 0.973
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.021 0.196 9.574 9.984 10.626
alpha1_hat 100 0 1.292 2.799 −5.567 0.953 9.036
alpha2_hat 100 0 4.988 0.341 4.140 4.967 5.810
alpha3_hat 100 0 1.001 0.047 0.903 1.002 1.120
gamma0_hat 100 0 1.091 0.296 0.172 1.088 1.665
gamma1_hat 100 0 −0.098 2.863 −5.344 0.146 6.711
gamma2_hat 100 0 0.989 0.043 0.897 0.989 1.111
gamma3_hat 100 0 0.994 0.046 0.875 0.990 1.097
theta_hat 100 0 0.061 0.067 −0.090 0.054 0.199
theta_hat_t_stats 100 0 0.909 1.014 −1.544 0.868 2.982
R2_demand 100 0 0.617 0.036 0.502 0.619 0.691
R2_supply 100 0 0.956 0.003 0.947 0.956 0.962
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.139 9.664 9.999 10.391
alpha1_hat 100 0 0.892 1.788 −4.114 0.834 4.986
alpha2_hat 100 0 5.011 0.207 4.459 5.035 5.508
alpha3_hat 100 0 0.999 0.029 0.933 0.997 1.060
gamma0_hat 100 0 1.055 0.195 0.551 1.059 1.599
gamma1_hat 100 0 0.116 2.033 −6.430 0.113 5.167
gamma2_hat 100 0 0.995 0.028 0.920 0.996 1.067
gamma3_hat 100 0 0.994 0.032 0.922 0.992 1.063
theta_hat 100 0 0.060 0.045 −0.053 0.063 0.207
theta_hat_t_stats 100 0 1.312 0.967 −1.225 1.387 4.180
R2_demand 100 0 0.619 0.023 0.550 0.624 0.662
R2_supply 100 0 0.955 0.002 0.948 0.955 0.961
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.082 9.776 9.995 10.227
alpha1_hat 100 0 0.978 1.209 −2.241 1.031 4.140
alpha2_hat 100 0 5.003 0.142 4.679 4.974 5.388
alpha3_hat 100 0 1.001 0.019 0.959 1.000 1.047
gamma0_hat 100 0 1.019 0.119 0.737 1.012 1.293
gamma1_hat 100 0 0.414 1.253 −2.735 0.604 3.096
gamma2_hat 100 0 0.999 0.018 0.954 0.998 1.047
gamma3_hat 100 0 0.999 0.019 0.926 1.001 1.047
theta_hat 100 0 0.059 0.025 0.006 0.056 0.122
theta_hat_t_stats 100 0 2.057 0.841 0.212 1.953 4.169
R2_demand 100 0 0.620 0.014 0.587 0.621 0.653
R2_supply 100 0 0.955 0.001 0.950 0.955 0.958

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.078 1.156 7.657 10.040 17.632
alpha1_hat 100 0 2.394 47.415 −116.140 1.858 116.686
alpha2_hat 100 0 20.067 5.963 8.920 19.537 49.105
alpha3_hat 100 0 1.028 0.243 0.532 0.999 2.502
gamma0_hat 100 0 2.937 0.912 −0.206 2.934 5.571
gamma1_hat 100 0 4.394 45.937 −85.423 5.066 177.419
gamma2_hat 100 0 0.784 0.160 0.394 0.800 1.104
gamma3_hat 100 0 0.779 0.164 0.482 0.773 1.371
theta_hat 100 0 −0.197 0.276 −1.358 −0.161 0.495
theta_hat_t_stats 100 0 −0.785 0.981 −3.183 −0.665 1.624
R2_demand 100 0 0.564 0.537 −4.527 0.655 0.827
R2_supply 100 0 0.969 0.009 0.917 0.969 0.983
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.682 8.803 9.929 12.743
alpha1_hat 100 0 2.544 33.319 −74.863 0.205 79.604
alpha2_hat 100 0 19.850 3.830 11.814 20.057 30.398
alpha3_hat 100 0 0.992 0.157 0.692 0.984 1.611
gamma0_hat 100 0 2.125 0.706 −0.080 2.160 3.545
gamma1_hat 100 0 −8.277 35.057 −128.621 −5.857 72.619
gamma2_hat 100 0 0.886 0.116 0.600 0.890 1.161
gamma3_hat 100 0 0.874 0.114 0.567 0.883 1.121
theta_hat 100 0 −0.038 0.203 −0.433 −0.050 0.611
theta_hat_t_stats 100 0 −0.307 1.123 −3.312 −0.284 2.416
R2_demand 100 0 0.618 0.129 −0.022 0.636 0.788
R2_supply 100 0 0.964 0.007 0.940 0.965 0.977
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.994 0.371 9.163 10.035 11.065
alpha1_hat 100 0 0.532 21.853 −67.392 −0.577 48.025
alpha2_hat 100 0 20.039 2.642 13.664 20.145 27.673
alpha3_hat 100 0 0.991 0.103 0.703 1.002 1.231
gamma0_hat 100 0 1.579 0.648 −0.396 1.620 2.777
gamma1_hat 100 0 −3.567 19.372 −65.107 −1.989 47.302
gamma2_hat 100 0 0.943 0.095 0.730 0.938 1.191
gamma3_hat 100 0 0.938 0.093 0.744 0.933 1.336
theta_hat 100 0 0.003 0.132 −0.288 −0.010 0.363
theta_hat_t_stats 100 0 −0.048 0.923 −2.355 −0.068 2.451
R2_demand 100 0 0.633 0.073 0.391 0.644 0.773
R2_supply 100 0 0.959 0.006 0.940 0.959 0.972
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.019 0.174 9.616 10.026 10.367
alpha1_hat 100 0 −0.831 10.322 −28.642 −1.953 26.857
alpha2_hat 100 0 20.251 1.177 16.694 20.211 23.084
alpha3_hat 100 0 1.006 0.041 0.912 1.012 1.103
gamma0_hat 100 0 1.037 0.298 0.296 1.033 1.940
gamma1_hat 100 0 −5.669 10.546 −31.365 −6.112 20.330
gamma2_hat 100 0 1.000 0.043 0.881 0.999 1.139
gamma3_hat 100 0 0.993 0.045 0.857 0.993 1.124
theta_hat 100 0 0.077 0.062 −0.065 0.078 0.211
theta_hat_t_stats 100 0 1.190 0.940 −1.156 1.217 3.214
R2_demand 100 0 0.634 0.028 0.545 0.635 0.696
R2_supply 100 0 0.954 0.004 0.941 0.954 0.962
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.137 9.610 10.001 10.327
alpha1_hat 100 0 1.356 8.204 −15.540 0.259 21.595
alpha2_hat 100 0 19.974 0.969 18.050 19.986 22.431
alpha3_hat 100 0 1.004 0.032 0.889 1.004 1.073
gamma0_hat 100 0 1.038 0.191 0.468 1.049 1.391
gamma1_hat 100 0 −1.267 6.793 −18.113 −1.247 16.619
gamma2_hat 100 0 0.997 0.031 0.928 0.997 1.087
gamma3_hat 100 0 0.994 0.032 0.911 0.991 1.074
theta_hat 100 0 0.057 0.041 −0.029 0.056 0.150
theta_hat_t_stats 100 0 1.250 0.890 −0.731 1.258 3.279
R2_demand 100 0 0.630 0.022 0.580 0.629 0.688
R2_supply 100 0 0.954 0.003 0.945 0.954 0.959
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.991 0.076 9.851 9.981 10.176
alpha1_hat 100 0 0.919 4.200 −9.902 0.690 14.379
alpha2_hat 100 0 19.977 0.462 18.729 20.022 21.168
alpha3_hat 100 0 1.000 0.017 0.952 1.000 1.040
gamma0_hat 100 0 1.015 0.132 0.642 1.024 1.327
gamma1_hat 100 0 −0.318 4.568 −11.722 −0.385 11.991
gamma2_hat 100 0 0.999 0.021 0.946 0.996 1.060
gamma3_hat 100 0 0.999 0.020 0.952 1.001 1.050
theta_hat 100 0 0.055 0.026 −0.030 0.055 0.119
theta_hat_t_stats 100 0 1.931 0.913 −1.128 1.958 4.174
R2_demand 100 0 0.631 0.013 0.602 0.633 0.659
R2_supply 100 0 0.953 0.001 0.950 0.953 0.957

\(\theta = 0.1\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.922 1.004 7.786 9.863 14.467
alpha1_hat 100 0 0.980 0.866 −0.802 0.867 3.118
alpha2_hat 100 0 0.097 0.084 −0.101 0.096 0.362
alpha3_hat 100 0 0.993 0.247 0.475 0.952 2.143
gamma0_hat 100 0 2.892 0.988 −0.253 2.980 5.471
gamma1_hat 100 0 0.010 7.095 −41.040 0.664 16.645
gamma2_hat 100 0 0.784 0.170 0.269 0.783 1.253
gamma3_hat 100 0 0.791 0.147 0.345 0.790 1.181
theta_hat 100 0 0.286 3.885 −6.646 −0.111 20.789
theta_hat_t_stats 100 0 −0.158 1.073 −2.478 −0.216 3.201
R2_demand 100 0 0.313 0.320 −1.342 0.399 0.705
R2_supply 100 0 0.983 0.004 0.974 0.984 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.982 0.793 8.536 9.913 14.488
alpha1_hat 100 0 0.983 0.706 −0.476 0.892 4.420
alpha2_hat 100 0 0.101 0.061 −0.055 0.101 0.364
alpha3_hat 100 0 0.998 0.183 0.695 0.974 2.060
gamma0_hat 100 0 2.222 0.666 −0.070 2.137 3.709
gamma1_hat 100 0 1.256 6.325 −18.097 0.790 50.485
gamma2_hat 100 0 0.861 0.106 0.613 0.867 1.254
gamma3_hat 100 0 0.866 0.130 0.528 0.856 1.176
theta_hat 100 0 −0.051 2.299 −12.842 −0.016 10.421
theta_hat_t_stats 100 0 0.041 1.041 −2.186 −0.041 2.428
R2_demand 100 0 0.279 0.311 −2.128 0.345 0.582
R2_supply 100 0 0.981 0.003 0.971 0.981 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.017 0.514 8.751 9.988 11.461
alpha1_hat 100 0 1.064 0.383 0.235 1.066 1.977
alpha2_hat 100 0 0.094 0.040 0.005 0.091 0.220
alpha3_hat 100 0 1.007 0.101 0.823 0.997 1.310
gamma0_hat 100 0 1.586 0.694 −0.451 1.564 3.133
gamma1_hat 100 0 0.717 1.565 −10.581 0.950 5.447
gamma2_hat 100 0 0.933 0.102 0.720 0.929 1.198
gamma3_hat 100 0 0.944 0.105 0.728 0.937 1.271
theta_hat 100 0 0.133 0.789 −2.257 0.079 5.603
theta_hat_t_stats 100 0 0.159 1.108 −2.455 0.209 2.976
R2_demand 100 0 0.310 0.134 −0.144 0.327 0.574
R2_supply 100 0 0.978 0.004 0.967 0.979 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.185 9.641 10.009 10.470
alpha1_hat 100 0 0.987 0.168 0.606 0.985 1.521
alpha2_hat 100 0 0.102 0.016 0.055 0.102 0.131
alpha3_hat 100 0 0.998 0.046 0.873 0.998 1.120
gamma0_hat 100 0 1.132 0.316 0.412 1.129 1.968
gamma1_hat 100 0 0.954 0.340 0.024 0.950 1.740
gamma2_hat 100 0 0.984 0.043 0.884 0.984 1.075
gamma3_hat 100 0 0.987 0.051 0.862 0.988 1.087
theta_hat 100 0 0.100 0.181 −0.285 0.090 0.626
theta_hat_t_stats 100 0 0.566 1.076 −1.891 0.563 3.805
R2_demand 100 0 0.313 0.051 0.186 0.323 0.403
R2_supply 100 0 0.976 0.002 0.971 0.976 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.150 9.650 10.002 10.424
alpha1_hat 100 0 0.987 0.123 0.569 0.993 1.312
alpha2_hat 100 0 0.101 0.012 0.072 0.101 0.125
alpha3_hat 100 0 1.001 0.036 0.910 1.001 1.110
gamma0_hat 100 0 1.033 0.220 0.418 1.038 1.522
gamma1_hat 100 0 0.914 0.255 0.107 0.922 1.475
gamma2_hat 100 0 0.997 0.033 0.918 0.992 1.096
gamma3_hat 100 0 0.997 0.032 0.916 1.000 1.074
theta_hat 100 0 0.138 0.131 −0.141 0.117 0.529
theta_hat_t_stats 100 0 1.113 1.029 −1.244 1.078 3.978
R2_demand 100 0 0.312 0.040 0.198 0.314 0.415
R2_supply 100 0 0.976 0.001 0.973 0.976 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.082 9.812 10.004 10.206
alpha1_hat 100 0 0.989 0.081 0.798 0.978 1.193
alpha2_hat 100 0 0.102 0.008 0.084 0.101 0.120
alpha3_hat 100 0 1.002 0.020 0.962 1.003 1.054
gamma0_hat 100 0 0.992 0.133 0.568 0.997 1.294
gamma1_hat 100 0 0.959 0.137 0.520 0.975 1.405
gamma2_hat 100 0 1.002 0.020 0.947 1.000 1.050
gamma3_hat 100 0 1.000 0.019 0.944 1.000 1.049
theta_hat 100 0 0.121 0.070 −0.075 0.118 0.337
theta_hat_t_stats 100 0 1.611 0.918 −0.885 1.539 4.520
R2_demand 100 0 0.312 0.026 0.243 0.314 0.360
R2_supply 100 0 0.975 0.001 0.973 0.975 0.978

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.978 1.100 7.673 9.744 14.551
alpha1_hat 100 0 1.068 2.043 −3.604 1.104 9.409
alpha2_hat 100 0 0.491 0.222 −0.065 0.470 1.043
alpha3_hat 100 0 0.979 0.220 0.433 0.949 1.703
gamma0_hat 100 0 2.983 1.028 0.200 3.098 5.262
gamma1_hat 100 0 18.483 181.127 −14.730 0.925 1811.410
gamma2_hat 100 0 0.782 0.169 0.354 0.770 1.271
gamma3_hat 100 0 0.781 0.167 0.356 0.768 1.186
theta_hat 100 0 −4.103 40.295 −402.988 −0.172 2.947
theta_hat_t_stats 100 0 −0.392 1.050 −2.365 −0.497 2.655
R2_demand 100 0 0.441 0.308 −1.486 0.501 0.798
R2_supply 100 0 0.978 0.005 0.961 0.979 0.989
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.726 8.691 9.844 12.232
alpha1_hat 100 0 1.088 1.283 −2.326 1.066 5.789
alpha2_hat 100 0 0.484 0.140 0.028 0.491 0.981
alpha3_hat 100 0 0.991 0.157 0.605 0.962 1.598
gamma0_hat 100 0 2.236 0.774 0.428 2.349 4.294
gamma1_hat 100 0 0.152 4.327 −40.126 0.575 4.938
gamma2_hat 100 0 0.854 0.130 0.565 0.847 1.316
gamma3_hat 100 0 0.855 0.125 0.456 0.853 1.171
theta_hat 100 0 0.091 0.897 −0.713 0.007 8.467
theta_hat_t_stats 100 0 −0.073 1.051 −3.166 0.016 2.879
R2_demand 100 0 0.469 0.192 −0.342 0.511 0.774
R2_supply 100 0 0.974 0.005 0.959 0.975 0.984
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.041 0.475 9.076 10.000 11.728
alpha1_hat 100 0 1.054 1.037 −1.468 0.926 3.057
alpha2_hat 100 0 0.500 0.112 0.204 0.497 0.780
alpha3_hat 100 0 1.010 0.110 0.765 1.016 1.246
gamma0_hat 100 0 1.570 0.703 −0.253 1.680 2.958
gamma1_hat 100 0 0.773 1.050 −1.318 0.770 3.983
gamma2_hat 100 0 0.941 0.108 0.737 0.935 1.208
gamma3_hat 100 0 0.918 0.096 0.715 0.896 1.153
theta_hat 100 0 0.054 0.208 −0.436 0.053 0.571
theta_hat_t_stats 100 0 0.216 1.089 −3.558 0.213 2.777
R2_demand 100 0 0.477 0.126 −0.315 0.500 0.643
R2_supply 100 0 0.970 0.005 0.954 0.971 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.042 0.185 9.652 10.034 10.520
alpha1_hat 100 0 0.969 0.399 −0.250 0.997 2.022
alpha2_hat 100 0 0.509 0.044 0.410 0.505 0.621
alpha3_hat 100 0 1.014 0.047 0.912 1.011 1.141
gamma0_hat 100 0 1.124 0.248 0.508 1.118 1.689
gamma1_hat 100 0 0.760 0.416 −0.294 0.751 1.676
gamma2_hat 100 0 0.984 0.041 0.905 0.982 1.097
gamma3_hat 100 0 0.990 0.040 0.911 0.989 1.108
theta_hat 100 0 0.123 0.073 −0.018 0.117 0.305
theta_hat_t_stats 100 0 1.452 0.844 −0.245 1.451 3.473
R2_demand 100 0 0.480 0.045 0.378 0.482 0.572
R2_supply 100 0 0.967 0.002 0.961 0.968 0.972
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.131 9.681 9.966 10.273
alpha1_hat 100 0 0.966 0.276 −0.048 0.946 1.588
alpha2_hat 100 0 0.500 0.029 0.443 0.499 0.601
alpha3_hat 100 0 0.997 0.031 0.923 0.993 1.096
gamma0_hat 100 0 1.046 0.206 0.419 1.059 1.545
gamma1_hat 100 0 0.840 0.352 −0.008 0.817 1.632
gamma2_hat 100 0 0.995 0.028 0.929 0.992 1.079
gamma3_hat 100 0 0.996 0.030 0.936 0.994 1.081
theta_hat 100 0 0.121 0.063 −0.009 0.113 0.297
theta_hat_t_stats 100 0 2.002 1.009 −0.139 1.904 4.502
R2_demand 100 0 0.494 0.031 0.424 0.492 0.594
R2_supply 100 0 0.967 0.002 0.960 0.967 0.971
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.087 9.848 9.996 10.264
alpha1_hat 100 0 1.028 0.182 0.705 1.010 1.504
alpha2_hat 100 0 0.498 0.019 0.447 0.501 0.534
alpha3_hat 100 0 0.999 0.019 0.961 0.995 1.057
gamma0_hat 100 0 1.041 0.121 0.703 1.046 1.313
gamma1_hat 100 0 0.920 0.223 0.353 0.907 1.352
gamma2_hat 100 0 0.995 0.019 0.955 0.996 1.054
gamma3_hat 100 0 0.996 0.019 0.950 0.996 1.046
theta_hat 100 0 0.108 0.039 0.039 0.107 0.214
theta_hat_t_stats 100 0 2.830 0.967 1.087 2.866 5.420
R2_demand 100 0 0.490 0.020 0.428 0.492 0.528
R2_supply 100 0 0.967 0.001 0.964 0.966 0.970

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.038 0.939 8.563 9.859 13.464
alpha1_hat 100 0 1.139 3.316 −7.905 0.999 9.244
alpha2_hat 100 0 0.998 0.364 0.259 0.969 1.958
alpha3_hat 100 0 1.000 0.203 0.591 0.988 1.603
gamma0_hat 100 0 2.814 0.928 −0.982 3.001 4.362
gamma1_hat 100 0 0.947 4.313 −13.915 1.424 22.212
gamma2_hat 100 0 0.799 0.154 0.501 0.785 1.293
gamma3_hat 100 0 0.802 0.173 0.265 0.811 1.372
theta_hat 100 0 −0.121 0.424 −1.644 −0.176 1.541
theta_hat_t_stats 100 0 −0.534 1.010 −3.560 −0.631 2.053
R2_demand 100 0 0.501 0.210 −0.175 0.550 0.766
R2_supply 100 0 0.975 0.006 0.949 0.976 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.699 8.589 10.008 12.681
alpha1_hat 100 0 0.966 2.437 −7.025 0.897 6.588
alpha2_hat 100 0 1.004 0.283 0.358 1.009 1.796
alpha3_hat 100 0 0.992 0.162 0.666 0.982 1.479
gamma0_hat 100 0 2.072 0.810 0.331 2.160 3.910
gamma1_hat 100 0 0.328 3.135 −11.179 0.558 7.827
gamma2_hat 100 0 0.884 0.132 0.517 0.882 1.157
gamma3_hat 100 0 0.874 0.112 0.585 0.884 1.124
theta_hat 100 0 0.020 0.320 −0.615 −0.002 1.418
theta_hat_t_stats 100 0 −0.056 1.088 −2.581 −0.011 2.371
R2_demand 100 0 0.535 0.142 0.009 0.559 0.772
R2_supply 100 0 0.970 0.006 0.955 0.971 0.982
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.052 0.470 9.294 9.975 11.816
alpha1_hat 100 0 0.985 1.597 −3.600 1.158 5.063
alpha2_hat 100 0 1.013 0.187 0.547 0.993 1.675
alpha3_hat 100 0 1.009 0.109 0.786 1.004 1.354
gamma0_hat 100 0 1.592 0.519 0.212 1.646 2.863
gamma1_hat 100 0 0.510 1.737 −4.169 0.554 4.488
gamma2_hat 100 0 0.936 0.085 0.757 0.926 1.178
gamma3_hat 100 0 0.932 0.083 0.747 0.930 1.180
theta_hat 100 0 0.064 0.169 −0.331 0.058 0.662
theta_hat_t_stats 100 0 0.352 0.968 −1.713 0.318 2.701
R2_demand 100 0 0.529 0.112 −0.065 0.552 0.714
R2_supply 100 0 0.967 0.005 0.947 0.967 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.198 9.512 9.995 10.631
alpha1_hat 100 0 1.035 0.685 −1.053 1.009 3.025
alpha2_hat 100 0 0.999 0.082 0.797 1.003 1.248
alpha3_hat 100 0 0.997 0.049 0.896 0.992 1.156
gamma0_hat 100 0 1.062 0.341 0.218 1.053 1.947
gamma1_hat 100 0 0.785 0.791 −0.974 0.730 2.507
gamma2_hat 100 0 0.993 0.050 0.868 0.998 1.106
gamma3_hat 100 0 0.986 0.048 0.858 0.984 1.134
theta_hat 100 0 0.113 0.083 −0.092 0.118 0.290
theta_hat_t_stats 100 0 1.436 1.048 −1.267 1.549 3.372
R2_demand 100 0 0.529 0.041 0.420 0.531 0.624
R2_supply 100 0 0.963 0.003 0.955 0.963 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.994 0.127 9.689 9.981 10.334
alpha1_hat 100 0 1.076 0.462 0.143 1.053 2.293
alpha2_hat 100 0 0.991 0.046 0.883 0.987 1.095
alpha3_hat 100 0 1.002 0.031 0.902 1.001 1.075
gamma0_hat 100 0 1.043 0.205 0.528 1.036 1.514
gamma1_hat 100 0 0.777 0.491 −0.914 0.806 1.927
gamma2_hat 100 0 0.996 0.029 0.922 0.998 1.077
gamma3_hat 100 0 0.993 0.033 0.920 0.992 1.066
theta_hat 100 0 0.114 0.051 −0.025 0.108 0.288
theta_hat_t_stats 100 0 2.105 0.905 −0.488 2.012 4.869
R2_demand 100 0 0.540 0.023 0.480 0.543 0.612
R2_supply 100 0 0.963 0.002 0.957 0.963 0.968
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.002 0.094 9.754 10.008 10.219
alpha1_hat 100 0 0.994 0.269 0.298 0.992 1.531
alpha2_hat 100 0 1.001 0.032 0.908 0.999 1.076
alpha3_hat 100 0 1.000 0.021 0.955 0.997 1.054
gamma0_hat 100 0 1.019 0.116 0.747 1.019 1.305
gamma1_hat 100 0 0.899 0.267 0.327 0.893 1.472
gamma2_hat 100 0 0.997 0.019 0.951 0.997 1.038
gamma3_hat 100 0 0.995 0.020 0.948 0.999 1.036
theta_hat 100 0 0.108 0.027 0.050 0.107 0.167
theta_hat_t_stats 100 0 3.209 0.770 1.479 3.228 4.868
R2_demand 100 0 0.539 0.018 0.494 0.538 0.586
R2_supply 100 0 0.963 0.001 0.960 0.963 0.966

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.721 8.034 9.826 12.227
alpha1_hat 100 0 0.732 12.626 −30.189 1.279 31.383
alpha2_hat 100 0 5.012 1.345 2.416 4.866 8.509
alpha3_hat 100 0 1.007 0.213 0.575 0.995 1.646
gamma0_hat 100 0 2.909 0.974 0.794 2.946 5.611
gamma1_hat 100 0 −1.424 12.045 −32.763 −0.614 22.017
gamma2_hat 100 0 0.785 0.162 0.493 0.768 1.260
gamma3_hat 100 0 0.788 0.164 0.437 0.793 1.217
theta_hat 100 0 −0.100 0.258 −0.641 −0.096 0.476
theta_hat_t_stats 100 0 −0.439 1.010 −3.013 −0.338 1.789
R2_demand 100 0 0.563 0.203 −0.275 0.602 0.819
R2_supply 100 0 0.973 0.006 0.954 0.974 0.985
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.938 0.567 8.736 9.990 11.303
alpha1_hat 100 0 1.609 10.753 −29.624 1.950 38.203
alpha2_hat 100 0 4.877 1.207 1.285 4.847 9.029
alpha3_hat 100 0 1.003 0.133 0.770 0.995 1.363
gamma0_hat 100 0 2.034 0.741 −0.306 2.138 4.336
gamma1_hat 100 0 −3.129 10.206 −54.722 −2.579 20.806
gamma2_hat 100 0 0.888 0.116 0.589 0.876 1.292
gamma3_hat 100 0 0.898 0.132 0.522 0.906 1.229
theta_hat 100 0 0.053 0.235 −0.380 0.017 1.222
theta_hat_t_stats 100 0 0.125 0.886 −2.836 0.083 2.072
R2_demand 100 0 0.591 0.112 0.243 0.613 0.799
R2_supply 100 0 0.967 0.006 0.943 0.968 0.979
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.030 0.447 8.912 9.959 11.050
alpha1_hat 100 0 0.800 6.976 −16.810 0.746 18.238
alpha2_hat 100 0 5.048 0.844 3.433 5.046 7.572
alpha3_hat 100 0 1.011 0.106 0.760 1.009 1.282
gamma0_hat 100 0 1.493 0.570 0.297 1.536 2.739
gamma1_hat 100 0 −0.586 7.242 −20.630 −0.893 19.034
gamma2_hat 100 0 0.949 0.090 0.760 0.949 1.171
gamma3_hat 100 0 0.949 0.094 0.754 0.946 1.187
theta_hat 100 0 0.070 0.160 −0.330 0.063 0.537
theta_hat_t_stats 100 0 0.416 1.007 −2.639 0.481 2.904
R2_demand 100 0 0.576 0.088 0.266 0.592 0.736
R2_supply 100 0 0.963 0.005 0.948 0.964 0.975
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.183 9.653 9.977 10.385
alpha1_hat 100 0 1.291 2.956 −5.136 1.479 7.205
alpha2_hat 100 0 4.973 0.360 4.186 4.986 5.969
alpha3_hat 100 0 1.008 0.041 0.921 1.010 1.142
gamma0_hat 100 0 1.016 0.290 0.230 1.004 1.657
gamma1_hat 100 0 −0.149 2.827 −6.893 0.005 5.369
gamma2_hat 100 0 0.999 0.045 0.880 0.997 1.096
gamma3_hat 100 0 0.999 0.040 0.915 1.002 1.102
theta_hat 100 0 0.121 0.067 −0.055 0.121 0.283
theta_hat_t_stats 100 0 1.722 0.900 −0.935 1.822 3.725
R2_demand 100 0 0.588 0.034 0.509 0.589 0.674
R2_supply 100 0 0.959 0.003 0.945 0.959 0.966
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.985 0.133 9.706 9.966 10.504
alpha1_hat 100 0 0.858 2.075 −4.874 0.888 6.281
alpha2_hat 100 0 4.998 0.252 4.389 4.956 5.689
alpha3_hat 100 0 0.998 0.028 0.949 0.999 1.086
gamma0_hat 100 0 1.034 0.208 0.437 1.029 1.627
gamma1_hat 100 0 −0.012 2.086 −4.701 0.026 5.679
gamma2_hat 100 0 0.996 0.030 0.925 0.998 1.073
gamma3_hat 100 0 0.999 0.037 0.908 0.999 1.088
theta_hat 100 0 0.116 0.052 −0.014 0.120 0.232
theta_hat_t_stats 100 0 2.405 1.037 −0.320 2.525 4.656
R2_demand 100 0 0.589 0.025 0.499 0.591 0.629
R2_supply 100 0 0.959 0.002 0.953 0.959 0.964
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.086 9.818 9.988 10.300
alpha1_hat 100 0 0.973 1.200 −1.703 0.997 3.804
alpha2_hat 100 0 5.003 0.145 4.683 5.007 5.338
alpha3_hat 100 0 1.001 0.020 0.954 0.999 1.047
gamma0_hat 100 0 1.020 0.132 0.763 1.016 1.308
gamma1_hat 100 0 0.739 1.314 −1.709 0.774 3.626
gamma2_hat 100 0 0.998 0.020 0.956 0.996 1.045
gamma3_hat 100 0 0.997 0.018 0.947 0.996 1.046
theta_hat 100 0 0.103 0.029 0.037 0.105 0.173
theta_hat_t_stats 100 0 3.417 0.927 1.246 3.520 5.639
R2_demand 100 0 0.586 0.019 0.531 0.586 0.626
R2_supply 100 0 0.958 0.001 0.955 0.958 0.962

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.009 0.849 8.005 9.985 12.852
alpha1_hat 100 0 −1.334 57.303 −166.001 −4.268 170.300
alpha2_hat 100 0 20.221 6.912 0.669 19.854 41.702
alpha3_hat 100 0 0.983 0.197 0.505 0.978 1.448
gamma0_hat 100 0 2.863 0.964 −0.398 2.959 5.120
gamma1_hat 100 0 −31.207 183.952 −1779.277 −7.554 101.065
gamma2_hat 100 0 0.805 0.160 0.321 0.810 1.230
gamma3_hat 100 0 0.803 0.153 0.492 0.802 1.149
theta_hat 100 0 0.042 1.043 −0.859 −0.097 9.877
theta_hat_t_stats 100 0 −0.312 1.056 −2.501 −0.412 2.327
R2_demand 100 0 0.585 0.161 −0.060 0.619 0.858
R2_supply 100 0 0.972 0.007 0.943 0.973 0.984
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.136 0.644 8.713 10.140 11.647
alpha1_hat 100 0 −0.463 41.128 −102.521 0.213 130.323
alpha2_hat 100 0 20.640 4.822 8.791 20.220 31.189
alpha3_hat 100 0 1.009 0.145 0.699 1.004 1.313
gamma0_hat 100 0 1.947 0.803 −0.751 1.899 3.948
gamma1_hat 100 0 −4.245 36.973 −90.244 0.119 85.974
gamma2_hat 100 0 0.901 0.135 0.580 0.904 1.350
gamma3_hat 100 0 0.884 0.128 0.611 0.886 1.224
theta_hat 100 0 0.002 0.205 −0.377 −0.028 0.484
theta_hat_t_stats 100 0 −0.053 1.046 −2.429 −0.151 2.352
R2_demand 100 0 0.563 0.146 0.035 0.586 0.793
R2_supply 100 0 0.967 0.006 0.944 0.967 0.979
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.963 0.437 9.013 9.902 11.204
alpha1_hat 100 0 −3.256 26.239 −73.104 −0.447 61.611
alpha2_hat 100 0 20.285 3.231 13.759 20.306 30.019
alpha3_hat 100 0 0.998 0.111 0.765 0.996 1.329
gamma0_hat 100 0 1.487 0.567 0.439 1.521 2.653
gamma1_hat 100 0 −6.876 22.782 −87.167 −7.047 73.582
gamma2_hat 100 0 0.947 0.085 0.730 0.939 1.122
gamma3_hat 100 0 0.953 0.095 0.738 0.954 1.203
theta_hat 100 0 0.082 0.137 −0.348 0.078 0.508
theta_hat_t_stats 100 0 0.512 0.917 −2.380 0.554 2.722
R2_demand 100 0 0.594 0.078 0.364 0.594 0.772
R2_supply 100 0 0.962 0.005 0.946 0.962 0.974
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.995 0.207 9.460 10.015 10.476
alpha1_hat 100 0 0.251 10.457 −23.319 2.316 21.523
alpha2_hat 100 0 20.076 1.288 16.737 19.948 23.231
alpha3_hat 100 0 0.999 0.049 0.851 0.997 1.119
gamma0_hat 100 0 1.006 0.337 0.285 0.979 1.771
gamma1_hat 100 0 −2.841 10.198 −23.601 −2.792 27.481
gamma2_hat 100 0 1.002 0.048 0.893 0.993 1.112
gamma3_hat 100 0 1.004 0.041 0.899 1.005 1.118
theta_hat 100 0 0.117 0.062 −0.024 0.125 0.238
theta_hat_t_stats 100 0 1.724 0.878 −0.350 1.841 3.418
R2_demand 100 0 0.595 0.039 0.461 0.595 0.667
R2_supply 100 0 0.958 0.003 0.948 0.957 0.966
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.142 9.640 9.988 10.326
alpha1_hat 100 0 0.453 8.219 −19.675 0.204 15.929
alpha2_hat 100 0 20.017 0.956 17.485 20.057 22.606
alpha3_hat 100 0 0.995 0.034 0.903 0.992 1.085
gamma0_hat 100 0 1.049 0.210 0.591 1.050 1.481
gamma1_hat 100 0 −2.200 8.785 −25.017 −2.620 21.102
gamma2_hat 100 0 0.990 0.029 0.921 0.991 1.064
gamma3_hat 100 0 0.999 0.030 0.934 0.998 1.080
theta_hat 100 0 0.113 0.051 −0.031 0.111 0.225
theta_hat_t_stats 100 0 2.390 0.997 −0.682 2.398 4.436
R2_demand 100 0 0.597 0.026 0.518 0.600 0.642
R2_supply 100 0 0.958 0.002 0.952 0.958 0.963
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.091 9.785 9.992 10.190
alpha1_hat 100 0 0.643 4.690 −10.223 0.769 11.030
alpha2_hat 100 0 20.054 0.586 18.926 20.051 21.517
alpha3_hat 100 0 1.001 0.020 0.945 1.003 1.045
gamma0_hat 100 0 1.027 0.147 0.690 1.029 1.374
gamma1_hat 100 0 0.749 4.711 −9.806 0.376 17.872
gamma2_hat 100 0 0.995 0.022 0.953 0.996 1.055
gamma3_hat 100 0 0.999 0.021 0.951 0.998 1.058
theta_hat 100 0 0.099 0.029 0.012 0.103 0.161
theta_hat_t_stats 100 0 3.371 0.954 0.398 3.474 5.403
R2_demand 100 0 0.594 0.017 0.556 0.596 0.628
R2_supply 100 0 0.957 0.002 0.953 0.957 0.960

\(\theta = 0.2\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.075 1.011 7.398 10.061 13.437
alpha1_hat 100 0 1.091 0.872 −0.569 1.004 5.993
alpha2_hat 100 0 0.095 0.088 −0.270 0.097 0.283
alpha3_hat 100 0 1.004 0.238 0.588 0.981 1.694
gamma0_hat 100 0 2.879 1.120 −0.875 3.002 5.823
gamma1_hat 100 0 2.126 15.596 −56.115 0.641 102.696
gamma2_hat 100 0 0.795 0.173 0.388 0.767 1.180
gamma3_hat 100 0 0.772 0.165 0.372 0.770 1.143
theta_hat 100 0 −0.849 8.934 −66.411 −0.067 30.588
theta_hat_t_stats 100 0 0.000 1.048 −2.132 −0.034 2.463
R2_demand 100 0 0.229 0.389 −1.627 0.339 0.609
R2_supply 100 0 0.984 0.004 0.974 0.984 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.926 0.661 8.653 9.862 11.727
alpha1_hat 100 0 0.794 0.635 −0.550 0.785 2.855
alpha2_hat 100 0 0.117 0.064 −0.036 0.113 0.357
alpha3_hat 100 0 1.000 0.145 0.676 0.988 1.460
gamma0_hat 100 0 2.033 0.742 0.439 2.023 3.761
gamma1_hat 100 0 0.737 1.862 −6.322 0.806 12.258
gamma2_hat 100 0 0.888 0.119 0.609 0.885 1.147
gamma3_hat 100 0 0.894 0.115 0.566 0.894 1.224
theta_hat 100 0 0.125 1.002 −6.217 0.057 3.777
theta_hat_t_stats 100 0 0.167 0.962 −2.034 0.097 2.294
R2_demand 100 0 0.269 0.185 −0.489 0.302 0.527
R2_supply 100 0 0.982 0.003 0.974 0.983 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.959 0.434 8.882 9.918 11.708
alpha1_hat 100 0 0.941 0.458 −0.074 0.949 2.347
alpha2_hat 100 0 0.104 0.043 −0.022 0.102 0.237
alpha3_hat 100 0 1.001 0.102 0.755 0.998 1.358
gamma0_hat 100 0 1.634 0.644 −0.249 1.580 3.173
gamma1_hat 100 0 0.834 1.274 −2.316 0.828 7.457
gamma2_hat 100 0 0.928 0.099 0.660 0.919 1.195
gamma3_hat 100 0 0.931 0.097 0.676 0.929 1.167
theta_hat 100 0 0.186 0.674 −3.012 0.168 2.364
theta_hat_t_stats 100 0 0.400 1.027 −2.723 0.375 2.559
R2_demand 100 0 0.264 0.137 −0.149 0.274 0.532
R2_supply 100 0 0.980 0.003 0.968 0.980 0.987
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.031 0.176 9.642 10.023 10.461
alpha1_hat 100 0 1.010 0.190 0.643 1.021 1.701
alpha2_hat 100 0 0.101 0.019 0.043 0.098 0.152
alpha3_hat 100 0 1.008 0.037 0.932 1.008 1.111
gamma0_hat 100 0 1.159 0.325 0.402 1.156 2.169
gamma1_hat 100 0 0.898 0.380 −0.209 0.910 1.728
gamma2_hat 100 0 0.979 0.049 0.836 0.979 1.093
gamma3_hat 100 0 0.980 0.045 0.870 0.984 1.106
theta_hat 100 0 0.223 0.196 −0.265 0.211 0.752
theta_hat_t_stats 100 0 1.164 0.947 −1.412 1.200 3.185
R2_demand 100 0 0.264 0.056 0.134 0.271 0.373
R2_supply 100 0 0.978 0.002 0.973 0.977 0.981
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.132 9.671 10.024 10.394
alpha1_hat 100 0 1.011 0.132 0.646 1.015 1.329
alpha2_hat 100 0 0.099 0.012 0.069 0.098 0.137
alpha3_hat 100 0 0.999 0.030 0.918 1.001 1.077
gamma0_hat 100 0 1.019 0.202 0.493 1.024 1.520
gamma1_hat 100 0 0.877 0.251 0.396 0.862 1.650
gamma2_hat 100 0 0.997 0.029 0.921 0.996 1.061
gamma3_hat 100 0 0.998 0.030 0.918 0.994 1.056
theta_hat 100 0 0.259 0.135 −0.127 0.253 0.537
theta_hat_t_stats 100 0 1.942 0.952 −1.111 1.920 3.797
R2_demand 100 0 0.276 0.042 0.161 0.276 0.375
R2_supply 100 0 0.977 0.001 0.974 0.977 0.980
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.995 0.092 9.780 10.005 10.230
alpha1_hat 100 0 0.988 0.087 0.790 0.990 1.212
alpha2_hat 100 0 0.101 0.008 0.081 0.101 0.120
alpha3_hat 100 0 0.999 0.021 0.942 0.998 1.052
gamma0_hat 100 0 1.007 0.115 0.715 1.004 1.270
gamma1_hat 100 0 0.938 0.152 0.592 0.944 1.256
gamma2_hat 100 0 1.001 0.018 0.959 1.000 1.047
gamma3_hat 100 0 0.999 0.018 0.955 0.998 1.038
theta_hat 100 0 0.229 0.072 0.072 0.227 0.411
theta_hat_t_stats 100 0 2.849 0.868 0.964 2.783 5.225
R2_demand 100 0 0.279 0.026 0.191 0.279 0.337
R2_supply 100 0 0.977 0.001 0.975 0.977 0.979

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.931 0.852 7.610 9.913 12.363
alpha1_hat 100 0 0.945 2.071 −5.565 0.842 6.752
alpha2_hat 100 0 0.494 0.218 −0.055 0.491 1.175
alpha3_hat 100 0 0.991 0.212 0.508 0.965 1.470
gamma0_hat 100 0 2.730 0.906 0.337 2.706 4.949
gamma1_hat 100 0 −0.168 4.818 −41.311 0.334 4.985
gamma2_hat 100 0 0.794 0.151 0.450 0.802 1.113
gamma3_hat 100 0 0.792 0.156 0.350 0.816 1.307
theta_hat 100 0 0.150 0.797 −0.721 0.058 6.350
theta_hat_t_stats 100 0 0.095 0.983 −2.438 0.169 2.315
R2_demand 100 0 0.366 0.272 −0.637 0.422 0.733
R2_supply 100 0 0.981 0.005 0.968 0.982 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.609 8.668 9.966 11.644
alpha1_hat 100 0 0.873 1.411 −2.676 0.841 4.144
alpha2_hat 100 0 0.514 0.145 0.207 0.512 1.042
alpha3_hat 100 0 1.013 0.139 0.662 1.010 1.360
gamma0_hat 100 0 2.129 0.893 −0.858 2.289 4.047
gamma1_hat 100 0 0.483 1.588 −4.342 0.465 5.354
gamma2_hat 100 0 0.878 0.135 0.655 0.867 1.264
gamma3_hat 100 0 0.870 0.135 0.597 0.860 1.208
theta_hat 100 0 0.115 0.309 −0.489 0.099 0.992
theta_hat_t_stats 100 0 0.365 1.014 −2.245 0.368 3.071
R2_demand 100 0 0.411 0.176 −0.371 0.446 0.713
R2_supply 100 0 0.977 0.004 0.963 0.977 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.971 0.392 9.083 9.932 11.142
alpha1_hat 100 0 0.983 1.008 −1.916 1.021 3.446
alpha2_hat 100 0 0.498 0.106 0.255 0.489 0.800
alpha3_hat 100 0 0.983 0.095 0.814 0.981 1.286
gamma0_hat 100 0 1.435 0.610 −0.478 1.446 3.056
gamma1_hat 100 0 0.520 1.315 −2.886 0.618 3.530
gamma2_hat 100 0 0.952 0.086 0.726 0.947 1.186
gamma3_hat 100 0 0.956 0.087 0.768 0.951 1.242
theta_hat 100 0 0.217 0.234 −0.298 0.206 0.968
theta_hat_t_stats 100 0 0.954 0.988 −1.847 0.940 3.169
R2_demand 100 0 0.426 0.108 0.137 0.451 0.643
R2_supply 100 0 0.973 0.004 0.960 0.974 0.983
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.028 0.204 9.613 10.024 10.608
alpha1_hat 100 0 1.011 0.439 0.159 1.018 2.029
alpha2_hat 100 0 0.503 0.053 0.384 0.510 0.622
alpha3_hat 100 0 1.010 0.046 0.915 1.012 1.112
gamma0_hat 100 0 1.074 0.312 0.328 1.102 2.039
gamma1_hat 100 0 0.736 0.581 −0.649 0.713 2.757
gamma2_hat 100 0 0.991 0.046 0.860 0.988 1.120
gamma3_hat 100 0 0.992 0.042 0.882 0.992 1.110
theta_hat 100 0 0.232 0.111 −0.177 0.216 0.527
theta_hat_t_stats 100 0 2.441 1.055 −1.913 2.391 5.832
R2_demand 100 0 0.425 0.045 0.296 0.425 0.511
R2_supply 100 0 0.970 0.003 0.962 0.970 0.977
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.135 9.705 9.999 10.503
alpha1_hat 100 0 1.015 0.304 0.296 1.053 1.931
alpha2_hat 100 0 0.499 0.031 0.434 0.495 0.592
alpha3_hat 100 0 1.002 0.030 0.923 1.002 1.124
gamma0_hat 100 0 1.026 0.212 0.532 1.033 1.526
gamma1_hat 100 0 0.796 0.364 0.020 0.817 1.711
gamma2_hat 100 0 0.996 0.030 0.930 0.995 1.064
gamma3_hat 100 0 0.997 0.030 0.906 0.997 1.071
theta_hat 100 0 0.231 0.065 0.045 0.242 0.337
theta_hat_t_stats 100 0 3.491 0.935 0.712 3.626 5.046
R2_demand 100 0 0.429 0.035 0.313 0.426 0.516
R2_supply 100 0 0.970 0.002 0.965 0.970 0.974
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.086 9.814 9.999 10.170
alpha1_hat 100 0 1.027 0.208 0.573 1.011 1.535
alpha2_hat 100 0 0.498 0.024 0.444 0.498 0.558
alpha3_hat 100 0 1.001 0.021 0.959 0.999 1.056
gamma0_hat 100 0 1.015 0.144 0.634 1.020 1.406
gamma1_hat 100 0 0.891 0.239 0.253 0.897 1.489
gamma2_hat 100 0 0.998 0.021 0.953 0.998 1.058
gamma3_hat 100 0 0.996 0.022 0.941 0.996 1.061
theta_hat 100 0 0.216 0.040 0.128 0.218 0.308
theta_hat_t_stats 100 0 5.245 0.926 3.098 5.296 7.701
R2_demand 100 0 0.427 0.022 0.380 0.429 0.481
R2_supply 100 0 0.970 0.001 0.967 0.970 0.973

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.040 1.033 8.325 9.983 14.154
alpha1_hat 100 0 0.732 3.527 −7.984 0.989 13.497
alpha2_hat 100 0 1.034 0.395 −0.040 0.982 2.165
alpha3_hat 100 0 1.018 0.228 0.547 1.015 2.066
gamma0_hat 100 0 2.806 0.903 0.666 2.748 4.949
gamma1_hat 100 0 −0.049 5.598 −38.026 0.477 15.426
gamma2_hat 100 0 0.805 0.160 0.407 0.807 1.224
gamma3_hat 100 0 0.816 0.169 0.458 0.793 1.311
theta_hat 100 0 0.031 0.510 −1.451 −0.023 2.762
theta_hat_t_stats 100 0 −0.021 1.158 −3.269 −0.077 3.272
R2_demand 100 0 0.442 0.271 −1.045 0.510 0.773
R2_supply 100 0 0.979 0.006 0.960 0.979 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.888 0.665 8.647 9.810 11.863
alpha1_hat 100 0 1.235 2.322 −4.598 1.410 5.977
alpha2_hat 100 0 0.950 0.277 0.235 0.969 1.804
alpha3_hat 100 0 0.979 0.162 0.634 0.967 1.363
gamma0_hat 100 0 2.169 0.833 0.123 2.095 4.067
gamma1_hat 100 0 0.105 3.702 −16.598 0.670 8.773
gamma2_hat 100 0 0.863 0.123 0.552 0.868 1.261
gamma3_hat 100 0 0.880 0.136 0.585 0.877 1.282
theta_hat 100 0 0.131 0.405 −0.622 0.078 2.151
theta_hat_t_stats 100 0 0.299 1.258 −2.383 0.305 3.226
R2_demand 100 0 0.474 0.143 −0.081 0.508 0.740
R2_supply 100 0 0.975 0.005 0.957 0.976 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.402 9.067 9.978 10.821
alpha1_hat 100 0 0.892 1.662 −2.511 0.818 5.197
alpha2_hat 100 0 1.010 0.174 0.651 1.021 1.376
alpha3_hat 100 0 1.008 0.093 0.737 1.010 1.240
gamma0_hat 100 0 1.517 0.639 0.021 1.582 2.956
gamma1_hat 100 0 0.650 1.603 −2.989 0.516 4.405
gamma2_hat 100 0 0.949 0.100 0.708 0.943 1.256
gamma3_hat 100 0 0.943 0.095 0.733 0.946 1.264
theta_hat 100 0 0.155 0.173 −0.212 0.147 0.508
theta_hat_t_stats 100 0 0.807 0.946 −1.174 0.819 2.895
R2_demand 100 0 0.475 0.105 0.002 0.487 0.678
R2_supply 100 0 0.971 0.005 0.956 0.972 0.980
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.988 0.171 9.578 9.975 10.431
alpha1_hat 100 0 0.821 0.715 −0.816 0.789 2.254
alpha2_hat 100 0 1.015 0.078 0.850 1.014 1.272
alpha3_hat 100 0 0.997 0.042 0.893 1.000 1.083
gamma0_hat 100 0 1.050 0.306 0.317 1.052 1.788
gamma1_hat 100 0 0.754 0.748 −0.994 0.768 2.583
gamma2_hat 100 0 0.994 0.044 0.871 0.993 1.093
gamma3_hat 100 0 0.994 0.043 0.889 0.994 1.104
theta_hat 100 0 0.217 0.077 0.038 0.224 0.380
theta_hat_t_stats 100 0 2.648 0.865 0.487 2.709 4.435
R2_demand 100 0 0.473 0.043 0.342 0.479 0.554
R2_supply 100 0 0.968 0.003 0.961 0.968 0.974
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.145 9.645 9.980 10.376
alpha1_hat 100 0 1.038 0.491 0.048 1.017 2.282
alpha2_hat 100 0 0.995 0.058 0.795 0.986 1.122
alpha3_hat 100 0 1.000 0.032 0.908 1.001 1.091
gamma0_hat 100 0 0.999 0.215 0.343 1.023 1.482
gamma1_hat 100 0 0.788 0.611 −0.696 0.839 2.254
gamma2_hat 100 0 1.000 0.034 0.926 0.996 1.079
gamma3_hat 100 0 0.997 0.031 0.906 1.000 1.107
theta_hat 100 0 0.220 0.062 0.066 0.218 0.390
theta_hat_t_stats 100 0 3.735 0.943 1.290 3.712 5.943
R2_demand 100 0 0.479 0.035 0.380 0.483 0.534
R2_supply 100 0 0.967 0.002 0.963 0.967 0.971
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.080 9.850 9.989 10.241
alpha1_hat 100 0 0.987 0.308 0.308 0.955 2.050
alpha2_hat 100 0 0.999 0.032 0.921 1.004 1.083
alpha3_hat 100 0 0.999 0.020 0.950 0.998 1.046
gamma0_hat 100 0 1.021 0.131 0.673 1.019 1.308
gamma1_hat 100 0 0.915 0.338 −0.032 0.936 1.649
gamma2_hat 100 0 0.999 0.020 0.962 0.997 1.046
gamma3_hat 100 0 0.996 0.020 0.938 0.997 1.047
theta_hat 100 0 0.205 0.034 0.118 0.204 0.296
theta_hat_t_stats 100 0 5.632 0.825 3.389 5.662 7.723
R2_demand 100 0 0.472 0.020 0.413 0.475 0.513
R2_supply 100 0 0.968 0.001 0.964 0.968 0.970

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.002 1.058 7.983 9.898 14.800
alpha1_hat 100 0 1.377 15.017 −48.230 0.894 38.341
alpha2_hat 100 0 4.940 1.918 0.544 4.849 15.191
alpha3_hat 100 0 1.006 0.242 0.572 0.991 2.311
gamma0_hat 100 0 2.639 1.233 −1.461 2.691 7.090
gamma1_hat 100 0 −5.666 22.525 −117.822 −0.387 28.257
gamma2_hat 100 0 0.819 0.199 0.173 0.800 1.507
gamma3_hat 100 0 0.832 0.186 0.267 0.810 1.368
theta_hat 100 0 0.114 0.557 −0.924 0.035 2.744
theta_hat_t_stats 100 0 0.052 1.136 −3.250 0.123 2.813
R2_demand 100 0 0.506 0.292 −1.126 0.570 0.816
R2_supply 100 0 0.975 0.006 0.956 0.976 0.987
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.019 0.671 8.679 10.062 12.231
alpha1_hat 100 0 2.664 10.353 −18.538 1.161 33.037
alpha2_hat 100 0 4.837 1.178 1.936 4.857 7.383
alpha3_hat 100 0 1.002 0.139 0.698 0.996 1.357
gamma0_hat 100 0 2.127 0.747 −0.080 2.126 4.044
gamma1_hat 100 0 −1.843 10.331 −42.356 −1.027 26.998
gamma2_hat 100 0 0.869 0.133 0.465 0.872 1.300
gamma3_hat 100 0 0.870 0.136 0.513 0.876 1.219
theta_hat 100 0 0.112 0.237 −0.339 0.096 1.154
theta_hat_t_stats 100 0 0.397 0.873 −1.644 0.489 2.782
R2_demand 100 0 0.507 0.151 −0.049 0.532 0.741
R2_supply 100 0 0.973 0.005 0.958 0.974 0.983
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.955 0.387 9.235 9.913 11.116
alpha1_hat 100 0 0.953 6.612 −20.606 1.011 18.557
alpha2_hat 100 0 4.949 0.771 3.250 4.930 7.343
alpha3_hat 100 0 0.980 0.087 0.739 0.973 1.348
gamma0_hat 100 0 1.523 0.665 −0.546 1.614 2.793
gamma1_hat 100 0 −1.884 6.949 −23.961 −1.222 16.818
gamma2_hat 100 0 0.943 0.101 0.754 0.933 1.219
gamma3_hat 100 0 0.938 0.095 0.723 0.938 1.192
theta_hat 100 0 0.191 0.169 −0.234 0.173 0.692
theta_hat_t_stats 100 0 1.087 0.923 −1.760 1.061 2.948
R2_demand 100 0 0.521 0.102 0.148 0.532 0.682
R2_supply 100 0 0.969 0.006 0.950 0.970 0.978
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.982 0.193 9.560 9.971 10.435
alpha1_hat 100 0 1.092 3.330 −6.629 0.974 7.706
alpha2_hat 100 0 4.972 0.378 4.148 5.007 5.920
alpha3_hat 100 0 0.992 0.049 0.891 0.986 1.130
gamma0_hat 100 0 1.028 0.327 0.107 1.020 1.731
gamma1_hat 100 0 −0.354 3.578 −8.907 −0.572 9.172
gamma2_hat 100 0 0.997 0.047 0.887 1.001 1.122
gamma3_hat 100 0 1.004 0.046 0.897 1.003 1.104
theta_hat 100 0 0.225 0.077 0.017 0.228 0.388
theta_hat_t_stats 100 0 2.969 0.938 0.255 3.125 5.071
R2_demand 100 0 0.521 0.039 0.432 0.517 0.609
R2_supply 100 0 0.965 0.003 0.957 0.965 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.999 0.134 9.692 10.002 10.278
alpha1_hat 100 0 1.050 2.303 −6.558 1.026 6.791
alpha2_hat 100 0 4.995 0.248 4.188 4.964 5.820
alpha3_hat 100 0 1.002 0.030 0.929 1.001 1.072
gamma0_hat 100 0 1.010 0.214 0.489 1.020 1.528
gamma1_hat 100 0 0.021 2.175 −5.899 0.050 4.856
gamma2_hat 100 0 1.000 0.033 0.907 0.997 1.096
gamma3_hat 100 0 0.992 0.031 0.913 0.997 1.067
theta_hat 100 0 0.218 0.051 0.099 0.221 0.353
theta_hat_t_stats 100 0 4.156 0.879 2.102 4.250 6.452
R2_demand 100 0 0.519 0.029 0.437 0.521 0.587
R2_supply 100 0 0.964 0.002 0.958 0.964 0.970
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.084 9.818 10.003 10.194
alpha1_hat 100 0 0.896 1.299 −2.603 1.018 4.193
alpha2_hat 100 0 5.009 0.169 4.558 5.006 5.406
alpha3_hat 100 0 1.001 0.019 0.951 1.001 1.055
gamma0_hat 100 0 1.006 0.123 0.676 1.017 1.341
gamma1_hat 100 0 0.677 1.371 −2.389 0.750 4.054
gamma2_hat 100 0 0.999 0.020 0.954 0.998 1.051
gamma3_hat 100 0 0.997 0.019 0.957 0.998 1.037
theta_hat 100 0 0.206 0.034 0.114 0.205 0.313
theta_hat_t_stats 100 0 6.296 0.929 3.542 6.268 8.897
R2_demand 100 0 0.519 0.017 0.475 0.520 0.551
R2_supply 100 0 0.964 0.001 0.962 0.964 0.967

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.907 0.938 7.866 9.809 12.850
alpha1_hat 100 0 1.370 51.562 −120.993 −5.489 170.972
alpha2_hat 100 0 19.581 6.126 3.832 19.461 40.350
alpha3_hat 100 0 0.994 0.232 0.524 0.955 1.846
gamma0_hat 100 0 2.782 1.020 −0.130 2.758 5.169
gamma1_hat 100 0 −6.241 71.822 −372.320 1.326 118.587
gamma2_hat 100 0 0.799 0.166 0.402 0.802 1.334
gamma3_hat 100 0 0.808 0.163 0.510 0.788 1.247
theta_hat 100 0 0.000 0.397 −0.679 −0.079 2.100
theta_hat_t_stats 100 0 −0.192 1.040 −2.780 −0.294 2.484
R2_demand 100 0 0.504 0.230 −0.385 0.553 0.813
R2_supply 100 0 0.975 0.006 0.958 0.976 0.987
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.964 0.590 8.849 9.864 12.076
alpha1_hat 100 0 −6.842 38.993 −106.509 −8.018 94.391
alpha2_hat 100 0 20.716 4.666 11.087 20.285 34.440
alpha3_hat 100 0 0.994 0.129 0.703 0.993 1.496
gamma0_hat 100 0 2.064 0.860 −0.807 2.162 3.949
gamma1_hat 100 0 −6.651 38.958 −137.241 −5.655 148.931
gamma2_hat 100 0 0.885 0.135 0.576 0.888 1.293
gamma3_hat 100 0 0.863 0.129 0.546 0.878 1.236
theta_hat 100 0 0.098 0.223 −0.574 0.078 0.722
theta_hat_t_stats 100 0 0.404 0.959 −2.281 0.360 2.534
R2_demand 100 0 0.526 0.168 −0.418 0.558 0.786
R2_supply 100 0 0.972 0.006 0.952 0.973 0.981
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.041 0.450 9.145 10.047 11.300
alpha1_hat 100 0 3.898 32.444 −76.175 2.590 80.157
alpha2_hat 100 0 19.927 3.680 12.924 19.748 32.442
alpha3_hat 100 0 1.025 0.102 0.792 1.028 1.270
gamma0_hat 100 0 1.611 0.562 0.131 1.690 3.150
gamma1_hat 100 0 −11.447 25.385 −72.824 −11.050 45.139
gamma2_hat 100 0 0.928 0.085 0.652 0.915 1.207
gamma3_hat 100 0 0.939 0.091 0.716 0.936 1.171
theta_hat 100 0 0.184 0.145 −0.124 0.170 0.559
theta_hat_t_stats 100 0 1.092 0.796 −0.992 1.093 2.576
R2_demand 100 0 0.517 0.094 0.251 0.529 0.722
R2_supply 100 0 0.968 0.005 0.954 0.969 0.977
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.986 0.172 9.642 9.959 10.463
alpha1_hat 100 0 1.846 13.554 −32.313 1.997 35.088
alpha2_hat 100 0 19.849 1.481 16.789 19.801 23.422
alpha3_hat 100 0 0.995 0.044 0.904 0.992 1.124
gamma0_hat 100 0 1.119 0.271 0.221 1.121 1.718
gamma1_hat 100 0 −5.323 12.637 −34.809 −6.603 30.998
gamma2_hat 100 0 0.988 0.039 0.903 0.982 1.095
gamma3_hat 100 0 0.989 0.045 0.892 0.986 1.126
theta_hat 100 0 0.217 0.074 0.051 0.228 0.388
theta_hat_t_stats 100 0 2.947 0.942 0.767 3.098 5.131
R2_demand 100 0 0.531 0.041 0.385 0.536 0.609
R2_supply 100 0 0.965 0.002 0.955 0.965 0.970
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.154 9.663 9.990 10.370
alpha1_hat 100 0 2.484 8.549 −26.399 3.687 21.425
alpha2_hat 100 0 19.802 1.119 17.548 19.748 23.647
alpha3_hat 100 0 0.999 0.034 0.918 1.000 1.075
gamma0_hat 100 0 1.024 0.191 0.566 1.011 1.494
gamma1_hat 100 0 −2.420 9.483 −27.181 −3.119 18.414
gamma2_hat 100 0 0.999 0.029 0.916 0.998 1.085
gamma3_hat 100 0 0.994 0.031 0.925 0.996 1.060
theta_hat 100 0 0.214 0.053 0.090 0.209 0.344
theta_hat_t_stats 100 0 4.118 0.887 1.869 4.080 5.951
R2_demand 100 0 0.530 0.029 0.470 0.531 0.594
R2_supply 100 0 0.964 0.002 0.958 0.964 0.968
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.092 9.755 10.000 10.283
alpha1_hat 100 0 0.720 5.440 −13.112 0.422 12.155
alpha2_hat 100 0 20.050 0.645 18.084 20.056 21.485
alpha3_hat 100 0 1.003 0.022 0.949 1.003 1.072
gamma0_hat 100 0 0.996 0.133 0.641 0.998 1.282
gamma1_hat 100 0 −0.020 5.373 −13.535 0.016 9.925
gamma2_hat 100 0 0.999 0.019 0.958 0.999 1.053
gamma3_hat 100 0 1.001 0.020 0.959 1.001 1.060
theta_hat 100 0 0.206 0.032 0.131 0.203 0.295
theta_hat_t_stats 100 0 6.423 0.901 4.262 6.388 8.706
R2_demand 100 0 0.530 0.021 0.467 0.532 0.578
R2_supply 100 0 0.963 0.001 0.960 0.963 0.967

\(\theta = 0.33\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.101 0.903 7.916 10.022 12.702
alpha1_hat 100 0 1.053 1.063 −1.100 0.896 4.684
alpha2_hat 100 0 0.102 0.093 −0.171 0.098 0.323
alpha3_hat 100 0 1.003 0.219 0.569 0.991 1.623
gamma0_hat 100 0 3.050 0.971 −0.051 3.104 4.989
gamma1_hat 100 0 −0.328 10.076 −93.934 0.906 11.701
gamma2_hat 100 0 0.768 0.161 0.293 0.748 1.236
gamma3_hat 100 0 0.760 0.137 0.439 0.760 1.133
theta_hat 100 0 0.476 4.086 −5.673 −0.041 36.647
theta_hat_t_stats 100 0 −0.076 0.967 −2.664 −0.063 2.492
R2_demand 100 0 0.169 0.305 −0.935 0.247 0.609
R2_supply 100 0 0.986 0.004 0.975 0.986 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.942 0.645 8.511 9.888 11.803
alpha1_hat 100 0 1.035 0.659 −1.098 1.009 2.949
alpha2_hat 100 0 0.094 0.060 −0.039 0.092 0.305
alpha3_hat 100 0 1.003 0.158 0.666 0.989 1.558
gamma0_hat 100 0 2.150 0.882 −1.646 2.183 3.732
gamma1_hat 100 0 1.378 6.796 −15.452 0.796 51.772
gamma2_hat 100 0 0.872 0.139 0.605 0.845 1.320
gamma3_hat 100 0 0.877 0.144 0.587 0.853 1.424
theta_hat 100 0 −0.102 3.410 −26.026 0.169 8.604
theta_hat_t_stats 100 0 0.300 0.965 −2.071 0.363 2.250
R2_demand 100 0 0.245 0.153 −0.357 0.283 0.542
R2_supply 100 0 0.983 0.004 0.965 0.984 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.021 0.418 9.172 9.970 11.266
alpha1_hat 100 0 1.039 0.462 −0.247 0.990 2.050
alpha2_hat 100 0 0.097 0.042 −0.003 0.096 0.235
alpha3_hat 100 0 1.008 0.096 0.801 1.004 1.253
gamma0_hat 100 0 1.540 0.564 −0.050 1.607 2.841
gamma1_hat 100 0 0.541 1.599 −11.989 0.731 3.955
gamma2_hat 100 0 0.937 0.084 0.739 0.926 1.149
gamma3_hat 100 0 0.949 0.088 0.748 0.951 1.153
theta_hat 100 0 0.450 0.787 −1.252 0.331 6.558
theta_hat_t_stats 100 0 0.801 0.772 −0.915 0.735 2.460
R2_demand 100 0 0.230 0.134 −0.151 0.238 0.537
R2_supply 100 0 0.981 0.003 0.972 0.982 0.987
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.987 0.203 9.377 9.972 10.663
alpha1_hat 100 0 1.005 0.200 0.332 1.003 1.444
alpha2_hat 100 0 0.099 0.020 0.053 0.101 0.155
alpha3_hat 100 0 1.004 0.048 0.843 1.004 1.118
gamma0_hat 100 0 1.140 0.297 0.404 1.150 1.827
gamma1_hat 100 0 0.852 0.405 −0.353 0.846 1.758
gamma2_hat 100 0 0.981 0.049 0.852 0.981 1.106
gamma3_hat 100 0 0.994 0.037 0.914 0.992 1.092
theta_hat 100 0 0.383 0.212 −0.141 0.382 1.159
theta_hat_t_stats 100 0 1.876 0.974 −0.772 2.002 4.150
R2_demand 100 0 0.241 0.058 0.101 0.249 0.371
R2_supply 100 0 0.979 0.002 0.975 0.979 0.982
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.010 0.136 9.632 10.018 10.284
alpha1_hat 100 0 1.010 0.130 0.719 1.001 1.343
alpha2_hat 100 0 0.099 0.014 0.067 0.098 0.142
alpha3_hat 100 0 0.998 0.033 0.931 0.999 1.087
gamma0_hat 100 0 1.070 0.187 0.465 1.118 1.449
gamma1_hat 100 0 0.858 0.315 0.014 0.839 1.814
gamma2_hat 100 0 0.991 0.030 0.917 0.990 1.100
gamma3_hat 100 0 0.991 0.027 0.934 0.992 1.063
theta_hat 100 0 0.386 0.159 −0.132 0.379 0.861
theta_hat_t_stats 100 0 2.743 1.092 −0.998 2.760 5.375
R2_demand 100 0 0.230 0.042 0.126 0.223 0.333
R2_supply 100 0 0.979 0.001 0.975 0.979 0.981
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.996 0.094 9.756 10.009 10.186
alpha1_hat 100 0 0.984 0.093 0.700 0.981 1.208
alpha2_hat 100 0 0.101 0.009 0.079 0.101 0.121
alpha3_hat 100 0 1.001 0.021 0.948 1.002 1.060
gamma0_hat 100 0 1.012 0.132 0.627 1.009 1.283
gamma1_hat 100 0 0.989 0.175 0.510 0.993 1.437
gamma2_hat 100 0 0.998 0.019 0.950 1.000 1.048
gamma3_hat 100 0 0.998 0.019 0.957 0.997 1.041
theta_hat 100 0 0.335 0.090 0.066 0.327 0.570
theta_hat_t_stats 100 0 3.868 0.976 0.890 3.892 6.185
R2_demand 100 0 0.240 0.027 0.161 0.242 0.292
R2_supply 100 0 0.978 0.001 0.976 0.978 0.980

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.788 0.869 7.639 9.751 13.200
alpha1_hat 100 0 0.846 2.077 −3.651 0.938 6.905
alpha2_hat 100 0 0.484 0.233 −0.058 0.490 1.293
alpha3_hat 100 0 0.971 0.219 0.492 0.951 1.737
gamma0_hat 100 0 2.877 0.940 0.285 2.796 5.486
gamma1_hat 100 0 −0.944 21.296 −142.577 0.759 88.585
gamma2_hat 100 0 0.798 0.153 0.404 0.803 1.215
gamma3_hat 100 0 0.786 0.147 0.442 0.777 1.108
theta_hat 100 0 0.375 4.051 −13.291 0.030 27.103
theta_hat_t_stats 100 0 0.113 1.038 −3.937 0.066 2.469
R2_demand 100 0 0.366 0.273 −1.213 0.436 0.762
R2_supply 100 0 0.983 0.004 0.971 0.983 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.124 0.601 8.625 10.097 11.521
alpha1_hat 100 0 0.972 1.435 −3.407 0.881 5.203
alpha2_hat 100 0 0.523 0.162 0.116 0.529 0.973
alpha3_hat 100 0 1.044 0.148 0.716 1.051 1.519
gamma0_hat 100 0 2.179 0.700 0.521 2.290 4.102
gamma1_hat 100 0 0.683 1.792 −4.501 0.586 6.677
gamma2_hat 100 0 0.877 0.108 0.658 0.882 1.126
gamma3_hat 100 0 0.870 0.124 0.529 0.860 1.239
theta_hat 100 0 0.182 0.311 −0.618 0.170 1.398
theta_hat_t_stats 100 0 0.538 0.820 −1.567 0.700 2.075
R2_demand 100 0 0.314 0.194 −0.334 0.351 0.618
R2_supply 100 0 0.980 0.004 0.968 0.981 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.969 0.451 8.923 9.933 11.168
alpha1_hat 100 0 0.994 1.062 −2.433 0.846 4.063
alpha2_hat 100 0 0.496 0.109 0.221 0.488 0.786
alpha3_hat 100 0 0.999 0.116 0.726 1.009 1.273
gamma0_hat 100 0 1.713 0.535 0.128 1.717 2.975
gamma1_hat 100 0 0.585 1.522 −3.143 0.668 3.671
gamma2_hat 100 0 0.917 0.081 0.709 0.908 1.179
gamma3_hat 100 0 0.934 0.094 0.700 0.932 1.166
theta_hat 100 0 0.288 0.261 −0.173 0.259 0.995
theta_hat_t_stats 100 0 1.217 1.047 −0.813 1.158 3.946
R2_demand 100 0 0.357 0.115 0.000 0.379 0.556
R2_supply 100 0 0.977 0.004 0.969 0.977 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.989 0.216 9.527 9.985 10.570
alpha1_hat 100 0 0.921 0.428 −0.146 0.935 2.314
alpha2_hat 100 0 0.506 0.054 0.332 0.504 0.620
alpha3_hat 100 0 1.000 0.042 0.899 1.000 1.119
gamma0_hat 100 0 1.116 0.279 0.036 1.151 1.686
gamma1_hat 100 0 0.864 0.545 −0.288 0.824 2.054
gamma2_hat 100 0 0.989 0.041 0.888 0.993 1.109
gamma3_hat 100 0 0.989 0.042 0.881 0.985 1.085
theta_hat 100 0 0.334 0.105 0.084 0.334 0.617
theta_hat_t_stats 100 0 3.298 0.901 0.805 3.368 5.172
R2_demand 100 0 0.355 0.058 0.160 0.364 0.475
R2_supply 100 0 0.974 0.002 0.969 0.974 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.983 0.126 9.699 9.986 10.273
alpha1_hat 100 0 0.951 0.318 0.161 0.927 1.761
alpha2_hat 100 0 0.503 0.034 0.427 0.500 0.587
alpha3_hat 100 0 0.996 0.031 0.916 0.995 1.082
gamma0_hat 100 0 1.018 0.217 0.438 1.033 1.555
gamma1_hat 100 0 0.847 0.392 −0.129 0.863 1.694
gamma2_hat 100 0 1.001 0.030 0.926 1.001 1.090
gamma3_hat 100 0 0.996 0.032 0.918 0.998 1.069
theta_hat 100 0 0.353 0.080 0.174 0.350 0.577
theta_hat_t_stats 100 0 4.880 0.953 2.478 4.947 6.943
R2_demand 100 0 0.366 0.031 0.286 0.368 0.439
R2_supply 100 0 0.973 0.001 0.970 0.973 0.977
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.003 0.085 9.817 9.998 10.200
alpha1_hat 100 0 1.034 0.202 0.571 1.027 1.492
alpha2_hat 100 0 0.497 0.022 0.452 0.499 0.546
alpha3_hat 100 0 1.000 0.021 0.946 1.000 1.058
gamma0_hat 100 0 1.020 0.123 0.714 1.024 1.336
gamma1_hat 100 0 0.841 0.260 0.122 0.851 1.691
gamma2_hat 100 0 0.998 0.017 0.947 0.999 1.039
gamma3_hat 100 0 0.997 0.019 0.953 0.997 1.042
theta_hat 100 0 0.353 0.048 0.173 0.352 0.491
theta_hat_t_stats 100 0 7.786 0.940 3.828 7.718 9.819
R2_demand 100 0 0.359 0.021 0.309 0.361 0.424
R2_supply 100 0 0.973 0.001 0.971 0.973 0.976

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.881 0.911 7.719 9.780 12.297
alpha1_hat 100 0 0.820 3.832 −10.976 1.218 10.451
alpha2_hat 100 0 0.987 0.444 0.155 0.988 2.274
alpha3_hat 100 0 0.974 0.233 0.323 0.985 1.659
gamma0_hat 100 0 2.865 0.932 −0.045 2.755 5.161
gamma1_hat 100 0 −0.172 5.544 −15.292 −0.002 34.410
gamma2_hat 100 0 0.801 0.161 0.411 0.828 1.157
gamma3_hat 100 0 0.809 0.164 0.424 0.805 1.243
theta_hat 100 0 0.154 0.547 −2.567 0.125 1.698
theta_hat_t_stats 100 0 0.269 0.983 −2.299 0.313 2.433
R2_demand 100 0 0.433 0.229 −0.605 0.487 0.756
R2_supply 100 0 0.981 0.004 0.970 0.981 0.990
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.911 0.619 8.764 9.845 11.944
alpha1_hat 100 0 0.851 2.377 −4.596 0.806 6.231
alpha2_hat 100 0 0.991 0.288 0.353 0.978 1.800
alpha3_hat 100 0 0.999 0.152 0.641 0.992 1.480
gamma0_hat 100 0 2.078 0.864 −0.685 2.031 4.021
gamma1_hat 100 0 0.455 3.207 −8.076 1.032 7.723
gamma2_hat 100 0 0.880 0.129 0.583 0.878 1.271
gamma3_hat 100 0 0.868 0.141 0.544 0.859 1.291
theta_hat 100 0 0.219 0.344 −0.420 0.180 1.063
theta_hat_t_stats 100 0 0.694 1.069 −1.751 0.691 3.124
R2_demand 100 0 0.402 0.166 −0.199 0.436 0.642
R2_supply 100 0 0.978 0.004 0.966 0.978 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.991 0.399 9.100 9.926 10.992
alpha1_hat 100 0 0.978 1.600 −2.596 1.086 4.918
alpha2_hat 100 0 1.004 0.184 0.606 0.988 1.468
alpha3_hat 100 0 0.998 0.087 0.807 0.986 1.228
gamma0_hat 100 0 1.621 0.605 0.046 1.659 3.814
gamma1_hat 100 0 0.664 1.997 −7.424 0.659 5.592
gamma2_hat 100 0 0.924 0.092 0.610 0.926 1.159
gamma3_hat 100 0 0.931 0.091 0.674 0.915 1.200
theta_hat 100 0 0.268 0.216 −0.273 0.286 1.089
theta_hat_t_stats 100 0 1.289 0.929 −1.035 1.343 3.277
R2_demand 100 0 0.377 0.113 0.060 0.406 0.605
R2_supply 100 0 0.975 0.004 0.961 0.975 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.009 0.199 9.552 10.022 10.515
alpha1_hat 100 0 1.064 0.807 −1.336 1.050 3.026
alpha2_hat 100 0 0.997 0.092 0.774 0.993 1.273
alpha3_hat 100 0 1.006 0.052 0.901 1.006 1.140
gamma0_hat 100 0 1.079 0.270 0.348 1.070 1.625
gamma1_hat 100 0 0.590 0.923 −1.620 0.617 2.482
gamma2_hat 100 0 0.990 0.039 0.922 0.983 1.092
gamma3_hat 100 0 0.995 0.041 0.902 0.998 1.094
theta_hat 100 0 0.355 0.101 0.145 0.355 0.632
theta_hat_t_stats 100 0 3.849 0.925 1.802 3.835 6.046
R2_demand 100 0 0.392 0.048 0.264 0.393 0.500
R2_supply 100 0 0.972 0.002 0.965 0.972 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.983 0.130 9.671 9.990 10.252
alpha1_hat 100 0 1.070 0.629 −0.608 1.069 2.884
alpha2_hat 100 0 0.989 0.064 0.812 0.983 1.179
alpha3_hat 100 0 0.996 0.032 0.915 0.996 1.087
gamma0_hat 100 0 1.015 0.225 0.392 1.026 1.454
gamma1_hat 100 0 0.741 0.648 −0.773 0.799 2.350
gamma2_hat 100 0 0.998 0.034 0.937 0.998 1.085
gamma3_hat 100 0 0.998 0.037 0.931 0.992 1.091
theta_hat 100 0 0.354 0.065 0.200 0.351 0.494
theta_hat_t_stats 100 0 5.439 0.846 3.388 5.365 7.324
R2_demand 100 0 0.400 0.034 0.302 0.401 0.492
R2_supply 100 0 0.972 0.002 0.966 0.972 0.975
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.015 0.085 9.826 10.024 10.269
alpha1_hat 100 0 1.057 0.391 −0.222 1.003 2.088
alpha2_hat 100 0 0.997 0.043 0.904 0.998 1.133
alpha3_hat 100 0 1.003 0.020 0.961 1.002 1.064
gamma0_hat 100 0 1.013 0.130 0.662 1.021 1.383
gamma1_hat 100 0 0.852 0.398 0.004 0.844 1.710
gamma2_hat 100 0 0.999 0.019 0.949 0.999 1.045
gamma3_hat 100 0 0.999 0.023 0.946 1.000 1.056
theta_hat 100 0 0.341 0.040 0.244 0.340 0.440
theta_hat_t_stats 100 0 8.495 0.908 5.954 8.520 10.872
R2_demand 100 0 0.395 0.020 0.347 0.395 0.445
R2_supply 100 0 0.972 0.001 0.970 0.972 0.974

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.155 1.110 8.022 9.940 14.074
alpha1_hat 100 0 −0.743 17.362 −47.100 −1.777 52.841
alpha2_hat 100 0 5.352 1.980 0.878 5.258 10.056
alpha3_hat 100 0 1.028 0.251 0.492 0.996 2.140
gamma0_hat 100 0 3.200 0.909 0.037 3.191 5.759
gamma1_hat 100 0 −1.397 19.524 −88.836 0.319 59.714
gamma2_hat 100 0 0.730 0.152 0.430 0.740 1.112
gamma3_hat 100 0 0.778 0.165 0.291 0.771 1.220
theta_hat 100 0 0.087 0.437 −0.787 0.017 2.224
theta_hat_t_stats 100 0 0.063 0.938 −2.139 0.079 2.313
R2_demand 100 0 0.344 0.381 −1.444 0.452 0.757
R2_supply 100 0 0.980 0.005 0.968 0.981 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.034 0.656 8.761 9.962 12.255
alpha1_hat 100 0 1.650 12.396 −33.550 0.718 38.354
alpha2_hat 100 0 4.988 1.422 2.300 4.944 9.145
alpha3_hat 100 0 1.001 0.156 0.577 1.020 1.546
gamma0_hat 100 0 2.090 0.828 −0.652 2.157 3.900
gamma1_hat 100 0 −3.650 13.271 −36.075 −2.140 23.745
gamma2_hat 100 0 0.891 0.129 0.605 0.885 1.204
gamma3_hat 100 0 0.866 0.121 0.562 0.857 1.201
theta_hat 100 0 0.253 0.297 −0.265 0.219 1.243
theta_hat_t_stats 100 0 0.869 0.951 −1.571 1.003 2.459
R2_demand 100 0 0.424 0.158 −0.076 0.453 0.721
R2_supply 100 0 0.976 0.005 0.958 0.977 0.986
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.080 0.461 9.105 10.044 11.438
alpha1_hat 100 0 2.168 8.411 −18.957 2.191 21.653
alpha2_hat 100 0 4.968 0.909 2.360 4.973 7.329
alpha3_hat 100 0 1.003 0.094 0.807 0.992 1.269
gamma0_hat 100 0 1.552 0.582 0.117 1.603 2.665
gamma1_hat 100 0 −3.450 9.324 −34.725 −3.329 17.438
gamma2_hat 100 0 0.934 0.096 0.753 0.934 1.152
gamma3_hat 100 0 0.938 0.096 0.704 0.933 1.177
theta_hat 100 0 0.336 0.223 −0.159 0.307 1.191
theta_hat_t_stats 100 0 1.749 0.932 −0.958 1.886 4.108
R2_demand 100 0 0.424 0.107 0.069 0.441 0.604
R2_supply 100 0 0.973 0.004 0.965 0.974 0.981
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.017 0.207 9.502 10.010 10.834
alpha1_hat 100 0 1.106 3.724 −11.821 0.780 11.378
alpha2_hat 100 0 5.009 0.401 4.036 5.032 6.438
alpha3_hat 100 0 0.998 0.050 0.890 1.000 1.130
gamma0_hat 100 0 1.048 0.325 0.246 1.043 1.708
gamma1_hat 100 0 −0.532 4.228 −11.339 −0.248 7.667
gamma2_hat 100 0 0.995 0.051 0.878 0.993 1.125
gamma3_hat 100 0 0.991 0.045 0.877 0.986 1.082
theta_hat 100 0 0.353 0.094 0.123 0.339 0.590
theta_hat_t_stats 100 0 4.240 0.953 1.727 4.229 6.446
R2_demand 100 0 0.435 0.049 0.297 0.443 0.561
R2_supply 100 0 0.970 0.002 0.960 0.970 0.974
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.146 9.661 10.001 10.377
alpha1_hat 100 0 0.880 2.928 −9.563 0.832 8.123
alpha2_hat 100 0 5.011 0.310 4.243 5.014 6.091
alpha3_hat 100 0 0.998 0.030 0.908 0.999 1.086
gamma0_hat 100 0 1.043 0.227 0.490 1.075 1.538
gamma1_hat 100 0 −0.058 2.869 −7.891 0.000 6.226
gamma2_hat 100 0 0.996 0.034 0.900 1.000 1.083
gamma3_hat 100 0 0.996 0.032 0.931 0.993 1.095
theta_hat 100 0 0.344 0.070 0.201 0.339 0.492
theta_hat_t_stats 100 0 5.959 0.953 3.716 5.980 7.901
R2_demand 100 0 0.437 0.032 0.337 0.438 0.501
R2_supply 100 0 0.970 0.002 0.965 0.970 0.975
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.998 0.088 9.713 9.995 10.212
alpha1_hat 100 0 0.987 1.683 −3.239 0.900 5.071
alpha2_hat 100 0 4.998 0.194 4.540 4.998 5.521
alpha3_hat 100 0 1.001 0.020 0.962 1.003 1.053
gamma0_hat 100 0 1.024 0.154 0.682 1.017 1.523
gamma1_hat 100 0 0.618 1.553 −3.164 0.602 5.249
gamma2_hat 100 0 0.998 0.021 0.933 1.000 1.047
gamma3_hat 100 0 0.997 0.022 0.947 0.996 1.053
theta_hat 100 0 0.334 0.034 0.244 0.333 0.429
theta_hat_t_stats 100 0 9.217 0.766 7.115 9.357 10.854
R2_demand 100 0 0.439 0.023 0.391 0.439 0.495
R2_supply 100 0 0.969 0.001 0.966 0.969 0.972

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.012 1.402 7.889 9.766 20.673
alpha1_hat 100 0 7.425 85.038 −130.833 7.690 601.521
alpha2_hat 100 0 19.400 7.157 3.583 18.262 41.696
alpha3_hat 100 0 1.006 0.267 0.477 0.984 2.735
gamma0_hat 100 0 2.883 0.997 −0.236 2.871 5.377
gamma1_hat 100 0 −30.850 81.198 −391.959 −18.674 127.774
gamma2_hat 100 0 0.796 0.167 0.373 0.813 1.343
gamma3_hat 100 0 0.812 0.170 0.419 0.799 1.186
theta_hat 100 0 0.208 0.466 −0.685 0.126 2.103
theta_hat_t_stats 100 0 0.371 0.988 −2.155 0.416 2.601
R2_demand 100 0 0.371 0.875 −7.955 0.533 0.791
R2_supply 100 0 0.979 0.004 0.967 0.979 0.988
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.079 0.710 8.357 9.956 13.374
alpha1_hat 100 0 −5.827 50.208 −119.702 1.818 123.228
alpha2_hat 100 0 20.975 6.353 4.808 19.588 36.780
alpha3_hat 100 0 1.019 0.150 0.700 0.997 1.424
gamma0_hat 100 0 2.038 0.801 −0.825 2.119 3.676
gamma1_hat 100 0 −9.338 63.009 −277.901 −4.041 163.699
gamma2_hat 100 0 0.901 0.125 0.556 0.879 1.251
gamma3_hat 100 0 0.870 0.128 0.575 0.857 1.201
theta_hat 100 0 0.234 0.370 −0.621 0.166 2.218
theta_hat_t_stats 100 0 0.808 0.995 −1.592 0.815 4.356
R2_demand 100 0 0.415 0.203 −0.875 0.450 0.720
R2_supply 100 0 0.976 0.005 0.955 0.977 0.985
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.922 0.407 9.079 9.935 10.884
alpha1_hat 100 0 −1.508 28.924 −68.330 −3.907 68.355
alpha2_hat 100 0 19.881 3.396 11.564 19.740 28.292
alpha3_hat 100 0 0.982 0.101 0.768 0.984 1.289
gamma0_hat 100 0 1.622 0.599 0.045 1.676 2.842
gamma1_hat 100 0 −5.032 37.076 −107.607 −6.090 109.824
gamma2_hat 100 0 0.927 0.093 0.666 0.915 1.170
gamma3_hat 100 0 0.933 0.105 0.651 0.932 1.239
theta_hat 100 0 0.278 0.215 −0.353 0.261 1.143
theta_hat_t_stats 100 0 1.491 1.009 −1.826 1.463 3.428
R2_demand 100 0 0.446 0.109 0.103 0.464 0.666
R2_supply 100 0 0.973 0.004 0.962 0.974 0.979
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.014 0.213 9.553 10.005 10.726
alpha1_hat 100 0 −0.823 13.567 −28.407 −2.822 40.858
alpha2_hat 100 0 20.238 1.701 16.009 20.199 24.180
alpha3_hat 100 0 1.003 0.048 0.920 1.002 1.161
gamma0_hat 100 0 1.076 0.285 0.199 1.116 1.692
gamma1_hat 100 0 −4.559 14.751 −59.925 −3.706 28.596
gamma2_hat 100 0 0.991 0.046 0.898 0.997 1.109
gamma3_hat 100 0 1.000 0.042 0.912 0.996 1.116
theta_hat 100 0 0.347 0.086 0.153 0.346 0.685
theta_hat_t_stats 100 0 4.300 0.874 2.278 4.263 6.952
R2_demand 100 0 0.450 0.053 0.302 0.458 0.542
R2_supply 100 0 0.969 0.002 0.963 0.969 0.974
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.988 0.137 9.717 9.973 10.406
alpha1_hat 100 0 1.148 10.368 −31.475 0.356 30.391
alpha2_hat 100 0 19.959 1.206 17.236 19.892 23.249
alpha3_hat 100 0 0.999 0.031 0.925 0.999 1.086
gamma0_hat 100 0 0.990 0.195 0.552 0.971 1.579
gamma1_hat 100 0 −2.194 9.683 −29.166 −2.142 19.494
gamma2_hat 100 0 1.001 0.028 0.904 1.000 1.067
gamma3_hat 100 0 0.998 0.033 0.918 0.999 1.067
theta_hat 100 0 0.348 0.057 0.228 0.347 0.506
theta_hat_t_stats 100 0 6.092 0.835 4.292 5.991 8.040
R2_demand 100 0 0.446 0.035 0.350 0.447 0.527
R2_supply 100 0 0.969 0.001 0.966 0.969 0.973
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.098 9.752 10.008 10.213
alpha1_hat 100 0 1.032 6.687 −12.985 1.126 13.148
alpha2_hat 100 0 20.003 0.827 17.757 19.938 22.084
alpha3_hat 100 0 1.001 0.022 0.938 1.002 1.048
gamma0_hat 100 0 1.018 0.139 0.707 1.026 1.318
gamma1_hat 100 0 −1.843 6.898 −22.752 −1.159 16.108
gamma2_hat 100 0 0.998 0.021 0.959 0.999 1.049
gamma3_hat 100 0 0.999 0.020 0.954 1.000 1.045
theta_hat 100 0 0.342 0.042 0.213 0.335 0.489
theta_hat_t_stats 100 0 9.586 0.935 6.552 9.565 12.395
R2_demand 100 0 0.447 0.022 0.389 0.449 0.490
R2_supply 100 0 0.969 0.001 0.967 0.969 0.971

\(\theta = 0.5\)

\(\alpha_2 = 0.1\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.068 1.045 7.915 9.982 13.936
alpha1_hat 100 0 1.002 1.373 −1.912 1.049 7.658
alpha2_hat 100 0 0.104 0.129 −0.276 0.093 0.585
alpha3_hat 100 0 0.987 0.208 0.523 0.976 1.566
gamma0_hat 100 0 3.049 1.067 0.921 3.128 6.887
gamma1_hat 100 0 2.505 18.837 −17.852 0.968 182.637
gamma2_hat 100 0 0.766 0.179 0.311 0.749 1.108
gamma3_hat 100 0 0.800 0.144 0.399 0.805 1.159
theta_hat 100 0 −0.377 7.777 −72.026 0.070 15.912
theta_hat_t_stats 100 0 0.130 1.068 −2.878 0.113 2.575
R2_demand 100 0 0.125 0.446 −2.641 0.237 0.556
R2_supply 100 0 0.987 0.003 0.978 0.988 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.942 0.620 8.837 9.822 12.284
alpha1_hat 100 0 1.077 0.863 −0.684 1.102 5.283
alpha2_hat 100 0 0.089 0.076 −0.161 0.093 0.261
alpha3_hat 100 0 0.994 0.148 0.680 0.990 1.396
gamma0_hat 100 0 2.151 0.852 0.040 2.256 4.073
gamma1_hat 100 0 0.290 4.025 −24.751 0.682 12.127
gamma2_hat 100 0 0.867 0.138 0.527 0.860 1.188
gamma3_hat 100 0 0.881 0.116 0.648 0.878 1.175
theta_hat 100 0 0.670 2.089 −5.121 0.333 13.268
theta_hat_t_stats 100 0 0.565 1.047 −2.419 0.605 3.498
R2_demand 100 0 0.200 0.209 −0.678 0.235 0.613
R2_supply 100 0 0.985 0.003 0.972 0.986 0.990
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.972 0.468 8.912 9.949 10.966
alpha1_hat 100 0 0.955 0.510 −0.215 0.908 2.705
alpha2_hat 100 0 0.102 0.050 −0.020 0.098 0.219
alpha3_hat 100 0 0.983 0.101 0.773 0.981 1.267
gamma0_hat 100 0 1.668 0.585 0.409 1.738 3.122
gamma1_hat 100 0 0.861 2.417 −5.738 0.702 19.291
gamma2_hat 100 0 0.925 0.089 0.659 0.913 1.124
gamma3_hat 100 0 0.927 0.083 0.777 0.923 1.122
theta_hat 100 0 0.438 1.374 −9.967 0.513 5.035
theta_hat_t_stats 100 0 0.940 1.055 −2.898 1.075 3.728
R2_demand 100 0 0.188 0.151 −0.202 0.204 0.447
R2_supply 100 0 0.983 0.003 0.974 0.983 0.989
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.197 9.598 9.982 10.443
alpha1_hat 100 0 0.975 0.235 0.360 0.962 1.452
alpha2_hat 100 0 0.102 0.023 0.059 0.100 0.187
alpha3_hat 100 0 1.001 0.046 0.915 0.997 1.101
gamma0_hat 100 0 1.174 0.290 0.388 1.167 1.863
gamma1_hat 100 0 0.947 0.484 −0.375 0.927 2.100
gamma2_hat 100 0 0.982 0.043 0.889 0.981 1.108
gamma3_hat 100 0 0.978 0.048 0.865 0.979 1.125
theta_hat 100 0 0.490 0.262 −0.091 0.461 1.218
theta_hat_t_stats 100 0 2.204 0.993 −0.477 2.163 4.211
R2_demand 100 0 0.195 0.066 0.021 0.200 0.320
R2_supply 100 0 0.981 0.001 0.977 0.981 0.984
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.139 9.666 10.013 10.278
alpha1_hat 100 0 0.995 0.176 0.583 0.985 1.451
alpha2_hat 100 0 0.101 0.017 0.064 0.102 0.144
alpha3_hat 100 0 1.005 0.030 0.930 1.005 1.080
gamma0_hat 100 0 1.027 0.214 0.541 1.047 1.539
gamma1_hat 100 0 0.840 0.353 −0.096 0.893 1.441
gamma2_hat 100 0 0.998 0.034 0.927 0.997 1.073
gamma3_hat 100 0 0.994 0.031 0.926 0.993 1.099
theta_hat 100 0 0.572 0.176 0.228 0.553 1.023
theta_hat_t_stats 100 0 3.694 0.863 1.614 3.656 5.705
R2_demand 100 0 0.202 0.041 0.083 0.205 0.277
R2_supply 100 0 0.980 0.001 0.978 0.980 0.982
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.084 9.802 10.012 10.204
alpha1_hat 100 0 1.022 0.106 0.801 1.032 1.255
alpha2_hat 100 0 0.098 0.010 0.077 0.098 0.123
alpha3_hat 100 0 1.001 0.021 0.945 0.999 1.046
gamma0_hat 100 0 1.025 0.139 0.685 1.037 1.325
gamma1_hat 100 0 0.897 0.236 0.396 0.890 1.429
gamma2_hat 100 0 0.997 0.020 0.956 0.997 1.054
gamma3_hat 100 0 0.995 0.019 0.950 0.994 1.050
theta_hat 100 0 0.545 0.119 0.287 0.561 0.793
theta_hat_t_stats 100 0 5.573 1.012 3.279 5.605 7.761
R2_demand 100 0 0.200 0.028 0.140 0.198 0.277
R2_supply 100 0 0.980 0.001 0.978 0.980 0.982

\(\alpha_2 = 0.5\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.918 1.030 7.960 9.700 14.079
alpha1_hat 100 0 0.635 2.503 −4.577 0.383 7.811
alpha2_hat 100 0 0.522 0.295 −0.166 0.537 1.309
alpha3_hat 100 0 0.993 0.217 0.623 0.971 1.780
gamma0_hat 100 0 2.870 0.898 0.388 2.973 4.710
gamma1_hat 100 0 −1.060 14.170 −101.261 0.756 27.039
gamma2_hat 100 0 0.783 0.154 0.470 0.767 1.221
gamma3_hat 100 0 0.774 0.149 0.484 0.769 1.356
theta_hat 100 0 0.590 2.925 −5.001 0.158 23.862
theta_hat_t_stats 100 0 0.379 1.037 −2.416 0.363 2.735
R2_demand 100 0 0.262 0.375 −1.820 0.348 0.696
R2_supply 100 0 0.984 0.004 0.971 0.984 0.993
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.042 0.710 8.658 9.920 12.798
alpha1_hat 100 0 1.142 1.940 −3.704 0.923 7.108
alpha2_hat 100 0 0.494 0.197 −0.054 0.486 1.028
alpha3_hat 100 0 1.019 0.154 0.644 1.032 1.481
gamma0_hat 100 0 2.202 0.746 −0.142 2.233 3.695
gamma1_hat 100 0 0.288 5.557 −19.952 0.912 38.596
gamma2_hat 100 0 0.871 0.126 0.559 0.860 1.286
gamma3_hat 100 0 0.864 0.116 0.570 0.873 1.291
theta_hat 100 0 0.400 0.918 −5.588 0.319 3.396
theta_hat_t_stats 100 0 0.927 0.983 −2.089 0.931 2.871
R2_demand 100 0 0.260 0.243 −1.117 0.336 0.554
R2_supply 100 0 0.982 0.004 0.964 0.982 0.989
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.008 0.452 9.236 9.951 11.002
alpha1_hat 100 0 0.901 1.288 −2.563 0.945 4.009
alpha2_hat 100 0 0.511 0.126 0.169 0.517 0.786
alpha3_hat 100 0 0.992 0.093 0.735 0.992 1.356
gamma0_hat 100 0 1.617 0.599 0.238 1.554 3.069
gamma1_hat 100 0 0.557 1.828 −8.838 0.664 5.145
gamma2_hat 100 0 0.930 0.094 0.733 0.939 1.271
gamma3_hat 100 0 0.925 0.099 0.718 0.926 1.221
theta_hat 100 0 0.468 0.351 −0.176 0.437 2.075
theta_hat_t_stats 100 0 1.752 0.924 −0.354 1.844 3.630
R2_demand 100 0 0.270 0.137 −0.168 0.277 0.526
R2_supply 100 0 0.979 0.003 0.969 0.980 0.987
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.985 0.190 9.511 9.980 10.607
alpha1_hat 100 0 1.057 0.577 −0.562 1.106 2.106
alpha2_hat 100 0 0.492 0.063 0.370 0.482 0.652
alpha3_hat 100 0 0.997 0.045 0.884 0.996 1.102
gamma0_hat 100 0 1.081 0.337 0.045 1.102 2.166
gamma1_hat 100 0 0.654 0.756 −1.054 0.659 2.111
gamma2_hat 100 0 0.990 0.050 0.797 0.987 1.131
gamma3_hat 100 0 0.992 0.046 0.869 0.986 1.137
theta_hat 100 0 0.547 0.123 0.288 0.563 0.914
theta_hat_t_stats 100 0 4.658 0.879 2.677 4.746 7.057
R2_demand 100 0 0.292 0.056 0.164 0.300 0.415
R2_supply 100 0 0.977 0.002 0.972 0.977 0.982
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.997 0.154 9.672 9.988 10.373
alpha1_hat 100 0 0.951 0.370 0.116 0.988 2.276
alpha2_hat 100 0 0.505 0.040 0.409 0.505 0.618
alpha3_hat 100 0 1.001 0.034 0.914 0.997 1.070
gamma0_hat 100 0 1.000 0.229 0.470 1.005 1.445
gamma1_hat 100 0 0.830 0.522 −0.450 0.771 1.838
gamma2_hat 100 0 1.001 0.035 0.931 1.003 1.079
gamma3_hat 100 0 1.000 0.036 0.906 1.000 1.085
theta_hat 100 0 0.528 0.092 0.333 0.538 0.718
theta_hat_t_stats 100 0 6.607 0.939 4.465 6.728 9.102
R2_demand 100 0 0.286 0.046 0.149 0.289 0.386
R2_supply 100 0 0.977 0.001 0.973 0.977 0.979
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.001 0.085 9.719 10.004 10.156
alpha1_hat 100 0 1.011 0.213 0.549 1.025 1.559
alpha2_hat 100 0 0.499 0.024 0.432 0.501 0.555
alpha3_hat 100 0 0.998 0.019 0.940 0.999 1.048
gamma0_hat 100 0 1.003 0.146 0.521 1.007 1.290
gamma1_hat 100 0 0.956 0.341 −0.212 1.008 1.794
gamma2_hat 100 0 1.000 0.022 0.956 1.000 1.053
gamma3_hat 100 0 1.000 0.021 0.940 0.997 1.068
theta_hat 100 0 0.506 0.064 0.356 0.502 0.703
theta_hat_t_stats 100 0 10.026 1.026 7.377 9.952 13.464
R2_demand 100 0 0.287 0.024 0.233 0.286 0.368
R2_supply 100 0 0.976 0.001 0.974 0.976 0.978

\(\alpha_2 = 1.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.025 1.054 8.416 9.819 13.799
alpha1_hat 100 0 1.174 4.966 −12.688 0.825 25.156
alpha2_hat 100 0 0.989 0.588 −0.632 0.949 2.952
alpha3_hat 100 0 1.004 0.258 0.489 0.952 1.923
gamma0_hat 100 0 2.822 1.059 −0.520 2.882 5.524
gamma1_hat 100 0 −4.686 26.379 −203.212 1.205 17.270
gamma2_hat 100 0 0.811 0.174 0.304 0.794 1.318
gamma3_hat 100 0 0.793 0.159 0.446 0.793 1.268
theta_hat 100 0 0.770 2.738 −0.908 0.178 23.289
theta_hat_t_stats 100 0 0.500 1.008 −2.223 0.536 3.267
R2_demand 100 0 0.275 0.424 −2.122 0.397 0.744
R2_supply 100 0 0.984 0.004 0.971 0.984 0.992
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.014 0.683 8.583 9.864 12.234
alpha1_hat 100 0 0.899 2.964 −7.428 0.718 8.488
alpha2_hat 100 0 1.017 0.314 0.322 1.005 2.141
alpha3_hat 100 0 0.993 0.136 0.714 0.979 1.387
gamma0_hat 100 0 2.043 0.741 0.487 1.998 4.197
gamma1_hat 100 0 0.100 3.856 −12.010 1.170 6.557
gamma2_hat 100 0 0.888 0.121 0.615 0.905 1.157
gamma3_hat 100 0 0.883 0.115 0.614 0.879 1.164
theta_hat 100 0 0.397 0.387 −0.322 0.317 1.903
theta_hat_t_stats 100 0 1.186 0.941 −1.134 1.157 3.407
R2_demand 100 0 0.304 0.191 −0.249 0.329 0.608
R2_supply 100 0 0.981 0.003 0.971 0.982 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.052 0.488 8.847 10.022 11.306
alpha1_hat 100 0 1.198 2.177 −3.931 1.367 6.972
alpha2_hat 100 0 0.993 0.241 0.512 0.976 1.761
alpha3_hat 100 0 1.016 0.106 0.780 1.002 1.317
gamma0_hat 100 0 1.508 0.594 −0.358 1.580 2.746
gamma1_hat 100 0 0.013 2.722 −7.193 0.628 5.311
gamma2_hat 100 0 0.939 0.088 0.748 0.924 1.234
gamma3_hat 100 0 0.944 0.090 0.769 0.926 1.167
theta_hat 100 0 0.511 0.301 −0.102 0.470 1.428
theta_hat_t_stats 100 0 2.084 0.929 −0.529 2.062 4.203
R2_demand 100 0 0.293 0.138 −0.167 0.305 0.528
R2_supply 100 0 0.978 0.004 0.965 0.979 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.032 0.204 9.599 10.041 10.669
alpha1_hat 100 0 1.042 0.942 −1.293 1.059 3.488
alpha2_hat 100 0 1.005 0.111 0.657 1.021 1.303
alpha3_hat 100 0 1.004 0.041 0.910 1.006 1.118
gamma0_hat 100 0 1.009 0.327 0.150 1.060 1.912
gamma1_hat 100 0 0.651 1.112 −2.178 0.746 4.315
gamma2_hat 100 0 0.997 0.051 0.892 0.993 1.138
gamma3_hat 100 0 0.998 0.040 0.870 0.999 1.091
theta_hat 100 0 0.529 0.121 0.185 0.524 0.945
theta_hat_t_stats 100 0 5.103 0.866 2.109 5.107 6.784
R2_demand 100 0 0.313 0.053 0.145 0.312 0.413
R2_supply 100 0 0.975 0.002 0.969 0.975 0.980
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.032 0.154 9.640 10.025 10.500
alpha1_hat 100 0 1.070 0.659 −0.595 0.976 2.432
alpha2_hat 100 0 1.003 0.079 0.770 1.007 1.191
alpha3_hat 100 0 1.005 0.031 0.933 1.000 1.081
gamma0_hat 100 0 0.993 0.241 0.417 1.010 1.598
gamma1_hat 100 0 0.752 0.863 −1.277 0.728 2.790
gamma2_hat 100 0 1.004 0.034 0.921 1.002 1.075
gamma3_hat 100 0 1.000 0.031 0.911 1.002 1.082
theta_hat 100 0 0.518 0.083 0.272 0.516 0.694
theta_hat_t_stats 100 0 7.195 0.935 4.613 7.164 9.484
R2_demand 100 0 0.313 0.042 0.214 0.318 0.400
R2_supply 100 0 0.975 0.002 0.971 0.975 0.980
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.987 0.095 9.744 9.980 10.205
alpha1_hat 100 0 0.971 0.444 −0.180 0.945 2.023
alpha2_hat 100 0 0.998 0.049 0.888 0.999 1.115
alpha3_hat 100 0 0.996 0.021 0.948 0.993 1.054
gamma0_hat 100 0 1.025 0.139 0.714 0.999 1.386
gamma1_hat 100 0 0.923 0.428 −0.014 0.904 1.899
gamma2_hat 100 0 0.996 0.021 0.944 0.998 1.064
gamma3_hat 100 0 0.997 0.021 0.948 0.995 1.053
theta_hat 100 0 0.506 0.044 0.385 0.505 0.623
theta_hat_t_stats 100 0 11.252 0.750 9.289 11.281 12.920
R2_demand 100 0 0.320 0.025 0.255 0.319 0.366
R2_supply 100 0 0.975 0.001 0.973 0.975 0.977

\(\alpha_2 = 5.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.905 0.857 7.903 9.839 13.046
alpha1_hat 100 0 1.229 19.905 −56.477 −1.308 46.467
alpha2_hat 100 0 4.838 2.291 −0.843 5.057 11.967
alpha3_hat 100 0 0.994 0.178 0.653 0.978 1.503
gamma0_hat 100 0 2.697 1.091 −0.526 2.753 4.874
gamma1_hat 100 0 −7.803 161.793 −846.570 −2.044 1171.309
gamma2_hat 100 0 0.815 0.184 0.394 0.822 1.209
gamma3_hat 100 0 0.823 0.161 0.349 0.815 1.322
theta_hat 100 0 0.377 4.604 −34.915 0.324 21.124
theta_hat_t_stats 100 0 0.674 1.082 −2.379 0.964 2.865
R2_demand 100 0 0.355 0.287 −1.042 0.408 0.736
R2_supply 100 0 0.983 0.004 0.969 0.983 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.060 0.675 8.425 10.004 12.044
alpha1_hat 100 0 0.282 13.587 −33.200 −1.095 43.051
alpha2_hat 100 0 5.147 1.474 0.676 5.129 9.030
alpha3_hat 100 0 0.999 0.156 0.745 0.993 1.553
gamma0_hat 100 0 2.160 0.805 −0.636 2.280 4.232
gamma1_hat 100 0 −1.724 22.473 −183.848 0.780 28.859
gamma2_hat 100 0 0.869 0.132 0.541 0.858 1.239
gamma3_hat 100 0 0.864 0.122 0.573 0.867 1.102
theta_hat 100 0 0.369 0.455 −0.263 0.336 3.938
theta_hat_t_stats 100 0 1.124 0.820 −1.195 1.208 2.909
R2_demand 100 0 0.312 0.260 −0.988 0.370 0.586
R2_supply 100 0 0.980 0.004 0.966 0.980 0.986
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.960 0.473 8.789 9.966 11.243
alpha1_hat 100 0 −0.095 9.647 −19.686 0.279 20.684
alpha2_hat 100 0 5.058 1.106 2.841 4.852 8.631
alpha3_hat 100 0 0.994 0.114 0.770 0.985 1.251
gamma0_hat 100 0 1.495 0.706 −1.429 1.475 3.526
gamma1_hat 100 0 −2.369 11.519 −42.349 −0.872 21.932
gamma2_hat 100 0 0.941 0.105 0.676 0.938 1.304
gamma3_hat 100 0 0.945 0.115 0.713 0.914 1.297
theta_hat 100 0 0.496 0.253 −0.025 0.456 1.478
theta_hat_t_stats 100 0 2.369 0.932 −0.134 2.305 4.479
R2_demand 100 0 0.342 0.130 −0.090 0.361 0.561
R2_supply 100 0 0.977 0.004 0.959 0.977 0.985
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.016 0.184 9.555 10.003 10.435
alpha1_hat 100 0 1.571 3.895 −7.121 1.534 10.478
alpha2_hat 100 0 4.965 0.453 3.772 4.948 5.792
alpha3_hat 100 0 1.003 0.043 0.913 1.002 1.110
gamma0_hat 100 0 1.091 0.272 0.313 1.085 1.673
gamma1_hat 100 0 −1.295 4.697 −12.273 −1.650 10.414
gamma2_hat 100 0 0.991 0.040 0.900 0.994 1.120
gamma3_hat 100 0 0.985 0.042 0.874 0.984 1.101
theta_hat 100 0 0.529 0.098 0.253 0.531 0.715
theta_hat_t_stats 100 0 5.649 0.864 3.361 5.640 7.545
R2_demand 100 0 0.341 0.053 0.182 0.347 0.431
R2_supply 100 0 0.975 0.002 0.969 0.975 0.979
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.005 0.139 9.680 10.007 10.451
alpha1_hat 100 0 1.166 2.547 −3.731 0.902 6.745
alpha2_hat 100 0 4.994 0.341 4.223 5.031 5.811
alpha3_hat 100 0 1.004 0.034 0.927 1.005 1.089
gamma0_hat 100 0 1.029 0.201 0.529 1.035 1.626
gamma1_hat 100 0 −0.547 3.412 −10.827 −0.504 8.409
gamma2_hat 100 0 0.996 0.028 0.902 0.995 1.071
gamma3_hat 100 0 0.995 0.029 0.880 0.997 1.045
theta_hat 100 0 0.525 0.083 0.321 0.529 0.731
theta_hat_t_stats 100 0 7.982 0.974 5.001 8.028 10.732
R2_demand 100 0 0.344 0.042 0.237 0.346 0.452
R2_supply 100 0 0.974 0.001 0.970 0.974 0.977
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.000 0.085 9.800 10.005 10.220
alpha1_hat 100 0 0.678 1.526 −2.974 0.854 3.926
alpha2_hat 100 0 5.032 0.195 4.544 5.025 5.640
alpha3_hat 100 0 0.998 0.018 0.954 1.001 1.045
gamma0_hat 100 0 0.993 0.142 0.659 0.993 1.310
gamma1_hat 100 0 0.922 1.897 −4.592 0.726 4.979
gamma2_hat 100 0 1.001 0.021 0.951 1.000 1.051
gamma3_hat 100 0 1.000 0.019 0.961 0.997 1.048
theta_hat 100 0 0.503 0.045 0.394 0.505 0.645
theta_hat_t_stats 100 0 12.348 0.867 10.498 12.312 14.451
R2_demand 100 0 0.348 0.023 0.293 0.348 0.397
R2_supply 100 0 0.974 0.001 0.971 0.974 0.976

\(\alpha_2 = 20.0\)

\(n = 50\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.952 1.012 8.115 9.864 13.583
alpha1_hat 100 0 −6.081 81.140 −175.858 −5.806 224.524
alpha2_hat 100 0 20.447 8.570 −2.178 20.765 39.275
alpha3_hat 100 0 0.966 0.235 0.550 0.939 1.762
gamma0_hat 100 0 2.911 1.150 −1.406 3.069 5.667
gamma1_hat 100 0 −201.496 1479.822 −14641.636 −16.661 133.864
gamma2_hat 100 0 0.793 0.188 0.356 0.771 1.491
gamma3_hat 100 0 0.798 0.173 0.458 0.783 1.238
theta_hat 100 0 1.360 8.445 −0.668 0.271 83.274
theta_hat_t_stats 100 0 0.673 1.151 −2.623 0.777 3.126
R2_demand 100 0 0.319 0.389 −1.852 0.434 0.806
R2_supply 100 0 0.983 0.005 0.954 0.984 0.991
\(n = 100\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.919 0.624 8.551 9.892 12.815
alpha1_hat 100 0 3.180 50.544 −105.249 3.688 130.740
alpha2_hat 100 0 19.360 6.359 5.189 18.479 43.643
alpha3_hat 100 0 1.000 0.152 0.649 0.982 1.449
gamma0_hat 100 0 2.153 0.689 0.240 2.147 3.507
gamma1_hat 100 0 −20.328 61.971 −224.774 −7.449 91.478
gamma2_hat 100 0 0.873 0.111 0.634 0.867 1.208
gamma3_hat 100 0 0.871 0.121 0.614 0.857 1.168
theta_hat 100 0 0.438 0.365 −0.215 0.400 1.501
theta_hat_t_stats 100 0 1.387 0.972 −0.904 1.351 3.698
R2_demand 100 0 0.351 0.274 −1.872 0.380 0.676
R2_supply 100 0 0.980 0.004 0.968 0.980 0.987
\(n = 200\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.013 0.473 8.633 9.987 11.164
alpha1_hat 100 0 0.905 38.348 −127.127 7.208 71.531
alpha2_hat 100 0 20.126 4.806 9.769 19.467 38.683
alpha3_hat 100 0 1.013 0.100 0.800 1.018 1.314
gamma0_hat 100 0 1.522 0.653 −0.903 1.586 2.669
gamma1_hat 100 0 −14.090 39.473 −124.585 −11.299 63.808
gamma2_hat 100 0 0.934 0.103 0.701 0.938 1.214
gamma3_hat 100 0 0.946 0.104 0.723 0.943 1.264
theta_hat 100 0 0.505 0.266 0.001 0.475 1.438
theta_hat_t_stats 100 0 2.370 0.924 0.009 2.382 4.915
R2_demand 100 0 0.326 0.140 −0.064 0.359 0.547
R2_supply 100 0 0.977 0.004 0.963 0.977 0.984
\(n = 1000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.192 9.555 9.974 10.513
alpha1_hat 100 0 2.092 14.772 −39.193 3.266 45.123
alpha2_hat 100 0 19.835 1.710 14.689 19.769 23.925
alpha3_hat 100 0 1.002 0.042 0.916 0.996 1.100
gamma0_hat 100 0 1.089 0.288 0.195 1.083 1.999
gamma1_hat 100 0 −3.879 16.591 −46.964 −3.304 34.530
gamma2_hat 100 0 0.989 0.043 0.878 0.990 1.119
gamma3_hat 100 0 0.996 0.048 0.896 0.991 1.134
theta_hat 100 0 0.512 0.096 0.296 0.520 0.734
theta_hat_t_stats 100 0 5.585 0.840 3.557 5.620 7.244
R2_demand 100 0 0.356 0.048 0.230 0.362 0.464
R2_supply 100 0 0.974 0.002 0.968 0.974 0.978
\(n = 2000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 9.992 0.192 9.555 9.974 10.513
alpha1_hat 100 0 2.092 14.772 −39.193 3.266 45.123
alpha2_hat 100 0 19.835 1.710 14.689 19.769 23.925
alpha3_hat 100 0 1.002 0.042 0.916 0.996 1.100
gamma0_hat 100 0 1.089 0.288 0.195 1.083 1.999
gamma1_hat 100 0 −3.879 16.591 −46.964 −3.304 34.530
gamma2_hat 100 0 0.989 0.043 0.878 0.990 1.119
gamma3_hat 100 0 0.996 0.048 0.896 0.991 1.134
theta_hat 100 0 0.512 0.096 0.296 0.520 0.734
theta_hat_t_stats 100 0 5.585 0.840 3.557 5.620 7.244
R2_demand 100 0 0.356 0.048 0.230 0.362 0.464
R2_supply 100 0 0.974 0.002 0.968 0.974 0.978
\(n = 5000\)
Unique (#) Missing (%) Mean SD Min Median Max
group_id_k 100 0 50.500 29.011 1.000 50.500 100.000
alpha0_hat 100 0 10.007 0.089 9.789 10.003 10.203
alpha1_hat 100 0 0.465 6.336 −19.340 0.529 15.995
alpha2_hat 100 0 20.081 0.789 18.157 20.126 22.977
alpha3_hat 100 0 1.002 0.019 0.961 1.000 1.045
gamma0_hat 100 0 1.012 0.134 0.715 1.018 1.321
gamma1_hat 100 0 −1.800 7.541 −27.611 −1.720 13.695
gamma2_hat 100 0 0.999 0.020 0.952 0.999 1.051
gamma3_hat 100 0 0.999 0.021 0.948 0.998 1.053
theta_hat 100 0 0.511 0.046 0.430 0.510 0.640
theta_hat_t_stats 100 0 12.775 0.871 10.778 12.748 14.861
R2_demand 100 0 0.356 0.021 0.296 0.357 0.399
R2_supply 100 0 0.974 0.001 0.970 0.974 0.976

Plot

iv_polynomial

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## Warning: Removed 4 rows containing missing values (`geom_col()`).

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## Warning: Removed 1 rows containing missing values (`geom_col()`).

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